From d4a90dfb8d8a56012928a600ea2f6bd4758b51f6 Mon Sep 17 00:00:00 2001 From: Ferruccio Guidi Date: Tue, 7 May 2013 16:07:42 +0000 Subject: [PATCH] - partial commit: refactoring in the components before "computation" --- .../lambdadelta/basic_2/etc/cpr/cprs_ltpr.etc | 32 ++ .../lambdadelta/basic_2/etc/cpr/cprs_tpss.etc | 38 ++ .../lambdadelta/basic_2/reduction/lpr_cpr.ma | 356 ------------------ .../lambdadelta/basic_2/reduction/lpr_cpss.ma | 271 ------------- .../basic_2/reduction/lpr_ldrop.ma | 13 + .../lambdadelta/basic_2/reduction/lpr_lpr.ma | 333 +++++++++++++++- .../lambdadelta/basic_2/reduction/lpr_lpss.ma | 258 ++++++++++++- .../basic_2/substitution/lpss_cpss.ma | 144 ------- .../basic_2/substitution/lpss_ldrop.ma | 13 + .../basic_2/substitution/lpss_lpss.ma | 137 +++++++ .../lambdadelta/basic_2/unfold/lpqs_cpqs.ma | 228 ----------- .../lambdadelta/basic_2/unfold/lpqs_ldrop.ma | 13 + .../lambdadelta/basic_2/unfold/lpqs_lpqs.ma | 221 +++++++++++ .../lambdadelta/basic_2/web/basic_2_src.tbl | 4 +- 14 files changed, 1056 insertions(+), 1005 deletions(-) create mode 100644 matita/matita/contribs/lambdadelta/basic_2/etc/cpr/cprs_ltpr.etc create mode 100644 matita/matita/contribs/lambdadelta/basic_2/etc/cpr/cprs_tpss.etc delete mode 100644 matita/matita/contribs/lambdadelta/basic_2/reduction/lpr_cpr.ma delete mode 100644 matita/matita/contribs/lambdadelta/basic_2/reduction/lpr_cpss.ma diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/cpr/cprs_ltpr.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/cpr/cprs_ltpr.etc new file mode 100644 index 000000000..2682a7609 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/cpr/cprs_ltpr.etc @@ -0,0 +1,32 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/reducibility/cpr_ltpr.ma". +include "basic_2/computation/cprs.ma". + +(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************) + +(* Properties concerning parallel unfold on terms ***************************) + +(* Basic_1: was only: pr3_subst1 *) +lemma cprs_tpss_ltpr: ∀L1,T1,U1,d,e. L1 ⊢ T1 ▶* [d, e] U1 → + ∀L2. L1 ➡ L2 → ∀T2. L2 ⊢ T1 ➡* T2 → + ∃∃U2. L2 ⊢ U1 ➡* U2 & L2 ⊢ T2 ▶* [d, e] U2. +#L1 #T1 #U1 #d #e #HTU1 #L2 #HL12 #T2 #HT12 elim HT12 -T2 +[ #T2 #HT12 + elim (cpr_tpss_ltpr … HL12 … HT12 … HTU1) -L1 -T1 /3 width=3/ +| #T #T2 #_ #HT2 * #U #HU1 #HTU + elim (cpr_tpss_ltpr … HT2 … HTU) -L1 -T // /3 width=3/ +] +qed. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/cpr/cprs_tpss.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/cpr/cprs_tpss.etc new file mode 100644 index 000000000..0c575ae02 --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/cpr/cprs_tpss.etc @@ -0,0 +1,38 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +include "basic_2/reducibility/cpr_tpss.ma". +include "basic_2/computation/cprs.ma". + +(* CONTEXT-SENSITIVE PARALLEL COMPUTATION ON TERMS **************************) + +(* Properties on partial unfold for terms ***********************************) + +lemma cprs_tpss_trans: ∀L,T1,T. L ⊢ T1 ➡* T → + ∀T2,d,e. L ⊢ T ▶* [d, e] T2 → L ⊢ T1 ➡* T2. +#L #T1 #T #H @(cprs_ind … H) -T /2 width=3/ /3 width=5/ +qed. + +lemma cprs_tps_trans: ∀L,T1,T. L ⊢ T1 ➡* T → + ∀T2,d,e. L ⊢ T ▶ [d, e] T2 → L ⊢ T1 ➡* T2. +/3 width=5 by inj, cprs_tpss_trans/ qed. (**) (* auto too slow without trace *) + +lemma cprs_tpss_conf: ∀L,T0,T1. L ⊢ T0 ➡* T1 → + ∀T2,d,e. L ⊢ T0 ▶* [d, e] T2 → + ∃∃T. L ⊢ T1 ▶* [d, e] T & L ⊢ T2 ➡* T. +#L #T0 #T1 #H @(cprs_ind … H) -T1 /2 width=3/ +#T #T1 #_ #HT1 #IHT0 #T2 #d #e #HT02 +elim (IHT0 … HT02) -T0 #T0 #HT0 #HT20 +elim (cpr_tpss_conf … HT1 … HT0) -T /3 width=5/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/lpr_cpr.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/lpr_cpr.ma deleted file mode 100644 index c0f33584c..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/reduction/lpr_cpr.ma +++ /dev/null @@ -1,356 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/grammar/lpx_sn_lpx_sn.ma". -include "basic_2/relocation/fsup.ma". -include "basic_2/reduction/lpr_ldrop.ma". - -(* SN PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS *****************************) - -(* Main properties on context-sensitive parallel reduction for terms ********) - -fact cpr_conf_lpr_atom_atom: - ∀I,L1,L2. ∃∃T. L1 ⊢ ⓪{I} ➡ T & L2 ⊢ ⓪{I} ➡ T. -/2 width=3/ qed-. - -fact cpr_conf_lpr_atom_delta: - ∀L0,i. ( - ∀L,T.♯{L, T} < ♯{L0, #i} → - ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 → - ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 → - ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0 - ) → - ∀K0,V0. ⇩[O, i] L0 ≡ K0.ⓓV0 → - ∀V2. K0 ⊢ V0 ➡ V2 → ∀T2. ⇧[O, i + 1] V2 ≡ T2 → - ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ➡ L2 → - ∃∃T. L1 ⊢ #i ➡ T & L2 ⊢ T2 ➡ T. -#L0 #i #IH #K0 #V0 #HLK0 #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02 -elim (lpr_ldrop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1 -elim (lpr_inv_pair1 … H1) -H1 #K1 #V1 #HK01 #HV01 #H destruct -elim (lpr_ldrop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2 -elim (lpr_inv_pair1 … H2) -H2 #K2 #W2 #HK02 #_ #H destruct -lapply (ldrop_fwd_ldrop2 … HLK2) -W2 #HLK2 -lapply (ldrop_pair2_fwd_fw … HLK0 (#i)) -HLK0 #HLK0 -elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2 -elim (lift_total V 0 (i+1)) #T #HVT -lapply (cpr_lift … HV2 … HLK2 … HVT2 … HVT) -K2 -V2 /3 width=6/ -qed-. - -(* Basic_1: includes: pr0_delta_delta pr2_delta_delta *) -fact cpr_conf_lpr_delta_delta: - ∀L0,i. ( - ∀L,T.♯{L, T} < ♯{L0, #i} → - ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 → - ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 → - ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0 - ) → - ∀K0,V0. ⇩[O, i] L0 ≡ K0.ⓓV0 → - ∀V1. K0 ⊢ V0 ➡ V1 → ∀T1. ⇧[O, i + 1] V1 ≡ T1 → - ∀KX,VX. ⇩[O, i] L0 ≡ KX.ⓓVX → - ∀V2. KX ⊢ VX ➡ V2 → ∀T2. ⇧[O, i + 1] V2 ≡ T2 → - ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ➡ L2 → - ∃∃T. L1 ⊢ T1 ➡ T & L2 ⊢ T2 ➡ T. -#L0 #i #IH #K0 #V0 #HLK0 #V1 #HV01 #T1 #HVT1 -#KX #VX #H #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02 -lapply (ldrop_mono … H … HLK0) -H #H destruct -elim (lpr_ldrop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1 -elim (lpr_inv_pair1 … H1) -H1 #K1 #W1 #HK01 #_ #H destruct -lapply (ldrop_fwd_ldrop2 … HLK1) -W1 #HLK1 -elim (lpr_ldrop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2 -elim (lpr_inv_pair1 … H2) -H2 #K2 #W2 #HK02 #_ #H destruct -lapply (ldrop_fwd_ldrop2 … HLK2) -W2 #HLK2 -lapply (ldrop_pair2_fwd_fw … HLK0 (#i)) -HLK0 #HLK0 -elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2 -elim (lift_total V 0 (i+1)) #T #HVT -lapply (cpr_lift … HV1 … HLK1 … HVT1 … HVT) -K1 -V1 -lapply (cpr_lift … HV2 … HLK2 … HVT2 … HVT) -K2 -V2 -V /2 width=3/ -qed-. - -fact cpr_conf_lpr_bind_bind: - ∀a,I,L0,V0,T0. ( - ∀L,T.♯{L,T} < ♯{L0,ⓑ{a,I}V0.T0} → - ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 → - ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 → - ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0 - ) → - ∀V1. L0 ⊢ V0 ➡ V1 → ∀T1. L0.ⓑ{I}V0 ⊢ T0 ➡ T1 → - ∀V2. L0 ⊢ V0 ➡ V2 → ∀T2. L0.ⓑ{I}V0 ⊢ T0 ➡ T2 → - ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ➡ L2 → - ∃∃T. L1 ⊢ ⓑ{a,I}V1.T1 ➡ T & L2 ⊢ ⓑ{a,I}V2.T2 ➡ T. -#a #I #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01 -#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02 -elim (IH … HV01 … HV02 … HL01 … HL02) // -elim (IH … HT01 … HT02 (L1.ⓑ{I}V1) … (L2.ⓑ{I}V2)) -IH // /2 width=1/ /3 width=5/ -qed-. - -fact cpr_conf_lpr_bind_zeta: - ∀L0,V0,T0. ( - ∀L,T.♯{L,T} < ♯{L0,+ⓓV0.T0} → - ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 → - ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 → - ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0 - ) → - ∀V1. L0 ⊢ V0 ➡ V1 → ∀T1. L0.ⓓV0 ⊢ T0 ➡ T1 → - ∀T2. L0.ⓓV0 ⊢ T0 ➡ T2 → ∀X2. ⇧[O, 1] X2 ≡ T2 → - ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ➡ L2 → - ∃∃T. L1 ⊢ +ⓓV1.T1 ➡ T & L2 ⊢ X2 ➡ T. -#L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01 -#T2 #HT02 #X2 #HXT2 #L1 #HL01 #L2 #HL02 -elim (IH … HT01 … HT02 (L1.ⓓV1) … (L2.ⓓV1)) -IH -HT01 -HT02 // /2 width=1/ -L0 -V0 -T0 #T #HT1 #HT2 -elim (cpr_inv_lift1 … HT2 L2 … HXT2) -T2 /2 width=1/ /3 width=3/ -qed-. - -fact cpr_conf_lpr_zeta_zeta: - ∀L0,V0,T0. ( - ∀L,T.♯{L,T} < ♯{L0,+ⓓV0.T0} → - ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 → - ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 → - ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0 - ) → - ∀T1. L0.ⓓV0 ⊢ T0 ➡ T1 → ∀X1. ⇧[O, 1] X1 ≡ T1 → - ∀T2. L0.ⓓV0 ⊢ T0 ➡ T2 → ∀X2. ⇧[O, 1] X2 ≡ T2 → - ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ➡ L2 → - ∃∃T. L1 ⊢ X1 ➡ T & L2 ⊢ X2 ➡ T. -#L0 #V0 #T0 #IH #T1 #HT01 #X1 #HXT1 -#T2 #HT02 #X2 #HXT2 #L1 #HL01 #L2 #HL02 -elim (IH … HT01 … HT02 (L1.ⓓV0) … (L2.ⓓV0)) -IH -HT01 -HT02 // /2 width=1/ -L0 -T0 #T #HT1 #HT2 -elim (cpr_inv_lift1 … HT1 L1 … HXT1) -T1 /2 width=1/ #T1 #HT1 #HXT1 -elim (cpr_inv_lift1 … HT2 L2 … HXT2) -T2 /2 width=1/ #T2 #HT2 #HXT2 -lapply (lift_inj … HT2 … HT1) -T #H destruct /2 width=3/ -qed-. - -fact cpr_conf_lpr_flat_flat: - ∀I,L0,V0,T0. ( - ∀L,T.♯{L,T} < ♯{L0,ⓕ{I}V0.T0} → - ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 → - ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 → - ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0 - ) → - ∀V1. L0 ⊢ V0 ➡ V1 → ∀T1. L0 ⊢ T0 ➡ T1 → - ∀V2. L0 ⊢ V0 ➡ V2 → ∀T2. L0 ⊢ T0 ➡ T2 → - ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ➡ L2 → - ∃∃T. L1 ⊢ ⓕ{I}V1.T1 ➡ T & L2 ⊢ ⓕ{I}V2.T2 ➡ T. -#I #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01 -#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02 -elim (IH … HV01 … HV02 … HL01 … HL02) // -elim (IH … HT01 … HT02 … HL01 … HL02) // /3 width=5/ -qed-. - -fact cpr_conf_lpr_flat_tau: - ∀L0,V0,T0. ( - ∀L,T.♯{L,T} < ♯{L0,ⓝV0.T0} → - ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 → - ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 → - ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0 - ) → - ∀V1,T1. L0 ⊢ T0 ➡ T1 → ∀T2. L0 ⊢ T0 ➡ T2 → - ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ➡ L2 → - ∃∃T. L1 ⊢ ⓝV1.T1 ➡ T & L2 ⊢ T2 ➡ T. -#L0 #V0 #T0 #IH #V1 #T1 #HT01 -#T2 #HT02 #L1 #HL01 #L2 #HL02 -elim (IH … HT01 … HT02 … HL01 … HL02) // -L0 -V0 -T0 /3 width=3/ -qed-. - -fact cpr_conf_lpr_tau_tau: - ∀L0,V0,T0. ( - ∀L,T.♯{L,T} < ♯{L0,ⓝV0.T0} → - ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 → - ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 → - ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0 - ) → - ∀T1. L0 ⊢ T0 ➡ T1 → ∀T2. L0 ⊢ T0 ➡ T2 → - ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ➡ L2 → - ∃∃T. L1 ⊢ T1 ➡ T & L2 ⊢ T2 ➡ T. -#L0 #V0 #T0 #IH #T1 #HT01 -#T2 #HT02 #L1 #HL01 #L2 #HL02 -elim (IH … HT01 … HT02 … HL01 … HL02) // -L0 -V0 -T0 /2 width=3/ -qed-. - -fact cpr_conf_lpr_flat_beta: - ∀a,L0,V0,W0,T0. ( - ∀L,T.♯{L,T} < ♯{L0,ⓐV0.ⓛ{a}W0.T0} → - ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 → - ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 → - ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0 - ) → - ∀V1. L0 ⊢ V0 ➡ V1 → ∀T1. L0 ⊢ ⓛ{a}W0.T0 ➡ T1 → - ∀V2. L0 ⊢ V0 ➡ V2 → ∀T2. L0.ⓛW0 ⊢ T0 ➡ T2 → - ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ➡ L2 → - ∃∃T. L1 ⊢ ⓐV1.T1 ➡ T & L2 ⊢ ⓓ{a}V2.T2 ➡ T. -#a #L0 #V0 #W0 #T0 #IH #V1 #HV01 #X #H -#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02 -elim (cpr_inv_abst1 … H) -H #W1 #T1 #HW01 #HT01 #H destruct -elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1/ #V #HV1 #HV2 -elim (IH … HT01 … HT02 (L1.ⓛW1) … (L2.ⓛW1)) /2 width=1/ -L0 -V0 -W0 -T0 #T #HT1 #HT2 -lapply (cpr_lsubr_trans … HT2 (L2.ⓓV2) ?) -HT2 /2 width=1/ /3 width=5/ -qed-. - -(* Basic-1: includes: - pr0_cong_upsilon_refl pr0_cong_upsilon_zeta - pr0_cong_upsilon_cong pr0_cong_upsilon_delta -*) -fact cpr_conf_lpr_flat_theta: - ∀a,L0,V0,W0,T0. ( - ∀L,T.♯{L,T} < ♯{L0,ⓐV0.ⓓ{a}W0.T0} → - ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 → - ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 → - ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0 - ) → - ∀V1. L0 ⊢ V0 ➡ V1 → ∀T1. L0 ⊢ ⓓ{a}W0.T0 ➡ T1 → - ∀V2. L0 ⊢ V0 ➡ V2 → ∀U2. ⇧[O, 1] V2 ≡ U2 → - ∀W2. L0 ⊢ W0 ➡ W2 → ∀T2. L0.ⓓW0 ⊢ T0 ➡ T2 → - ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ➡ L2 → - ∃∃T. L1 ⊢ ⓐV1.T1 ➡ T & L2 ⊢ ⓓ{a}W2.ⓐU2.T2 ➡ T. -#a #L0 #V0 #W0 #T0 #IH #V1 #HV01 #X #H -#V2 #HV02 #U2 #HVU2 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02 -elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1/ #V #HV1 #HV2 -elim (lift_total V 0 1) #U #HVU -lapply (cpr_lift … HV2 (L2.ⓓW2) … HVU2 … HVU) -HVU2 /2 width=1/ #HU2 -elim (cpr_inv_abbr1 … H) -H * -[ #W1 #T1 #HW01 #HT01 #H destruct - elim (IH … HW01 … HW02 … HL01 … HL02) /2 width=1/ - elim (IH … HT01 … HT02 (L1.ⓓW1) … (L2.ⓓW2)) /2 width=1/ -L0 -V0 -W0 -T0 /4 width=7/ -| #T1 #HT01 #HXT1 #H destruct - elim (IH … HT01 … HT02 (L1.ⓓW2) … (L2.ⓓW2)) /2 width=1/ -L0 -V0 -W0 -T0 #T #HT1 #HT2 - elim (cpr_inv_lift1 … HT1 L1 … HXT1) -HXT1 /2 width=1/ #Y #HYT #HXY - @(ex2_intro … (ⓐV.Y)) /2 width=1/ /3 width=5/ (**) (* auto /4 width=9/ is too slow *) -] -qed-. - -fact cpr_conf_lpr_beta_beta: - ∀a,L0,V0,W0,T0. ( - ∀L,T.♯{L,T} < ♯{L0,ⓐV0.ⓛ{a}W0.T0} → - ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 → - ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 → - ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0 - ) → - ∀V1. L0 ⊢ V0 ➡ V1 → ∀T1. L0.ⓛW0 ⊢ T0 ➡ T1 → - ∀V2. L0 ⊢ V0 ➡ V2 → ∀T2. L0.ⓛW0 ⊢ T0 ➡ T2 → - ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ➡ L2 → - ∃∃T. L1 ⊢ ⓓ{a}V1.T1 ➡ T & L2 ⊢ ⓓ{a}V2.T2 ➡ T. -#a #L0 #V0 #W0 #T0 #IH #V1 #HV01 #T1 #HT01 -#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02 -elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1/ #V #HV1 #HV2 -elim (IH … HT01 … HT02 (L1.ⓛW0) … (L2.ⓛW0)) /2 width=1/ -L0 -V0 -T0 #T #HT1 #HT2 -lapply (cpr_lsubr_trans … HT1 (L1.ⓓV1) ?) -HT1 /2 width=1/ -lapply (cpr_lsubr_trans … HT2 (L2.ⓓV2) ?) -HT2 /2 width=1/ /3 width=5/ -qed-. - -(* Basic_1: was: pr0_upsilon_upsilon *) -fact cpr_conf_lpr_theta_theta: - ∀a,L0,V0,W0,T0. ( - ∀L,T.♯{L,T} < ♯{L0,ⓐV0.ⓓ{a}W0.T0} → - ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 → - ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 → - ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0 - ) → - ∀V1. L0 ⊢ V0 ➡ V1 → ∀U1. ⇧[O, 1] V1 ≡ U1 → - ∀W1. L0 ⊢ W0 ➡ W1 → ∀T1. L0.ⓓW0 ⊢ T0 ➡ T1 → - ∀V2. L0 ⊢ V0 ➡ V2 → ∀U2. ⇧[O, 1] V2 ≡ U2 → - ∀W2. L0 ⊢ W0 ➡ W2 → ∀T2. L0.ⓓW0 ⊢ T0 ➡ T2 → - ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ➡ L2 → - ∃∃T. L1 ⊢ ⓓ{a}W1.ⓐU1.T1 ➡ T & L2 ⊢ ⓓ{a}W2.ⓐU2.T2 ➡ T. -#a #L0 #V0 #W0 #T0 #IH #V1 #HV01 #U1 #HVU1 #W1 #HW01 #T1 #HT01 -#V2 #HV02 #U2 #HVU2 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02 -elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1/ #V #HV1 #HV2 -elim (IH … HW01 … HW02 … HL01 … HL02) /2 width=1/ -elim (IH … HT01 … HT02 (L1.ⓓW1) … (L2.ⓓW2)) /2 width=1/ -L0 -V0 -W0 -T0 -elim (lift_total V 0 1) #U #HVU -lapply (cpr_lift … HV1 (L1.ⓓW1) … HVU1 … HVU) -HVU1 /2 width=1/ -lapply (cpr_lift … HV2 (L2.ⓓW2) … HVU2 … HVU) -HVU2 /2 width=1/ /4 width=7/ -qed-. - -theorem cpr_conf_lpr: lpx_sn_confluent cpr cpr. -#L0 #T0 @(f2_ind … fw … L0 T0) -L0 -T0 #n #IH #L0 * [|*] -[ #I0 #Hn #T1 #H1 #T2 #H2 #L1 #HL01 #L2 #HL02 destruct - elim (cpr_inv_atom1 … H1) -H1 - elim (cpr_inv_atom1 … H2) -H2 - [ #H2 #H1 destruct - /2 width=1 by cpr_conf_lpr_atom_atom/ - | * #K0 #V0 #V2 #i2 #HLK0 #HV02 #HVT2 #H2 #H1 destruct - /3 width=10 by cpr_conf_lpr_atom_delta/ - | #H2 * #K0 #V0 #V1 #i1 #HLK0 #HV01 #HVT1 #H1 destruct - /4 width=10 by ex2_commute, cpr_conf_lpr_atom_delta/ - | * #X #Y #V2 #z #H #HV02 #HVT2 #H2 - * #K0 #V0 #V1 #i #HLK0 #HV01 #HVT1 #H1 destruct - /3 width=17 by cpr_conf_lpr_delta_delta/ - ] -| #a #I #V0 #T0 #Hn #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct - elim (cpr_inv_bind1 … H1) -H1 * - [ #V1 #T1 #HV01 #HT01 #H1 - | #T1 #HT01 #HXT1 #H11 #H12 - ] - elim (cpr_inv_bind1 … H2) -H2 * - [1,3: #V2 #T2 #HV02 #HT02 #H2 - |2,4: #T2 #HT02 #HXT2 #H21 #H22 - ] destruct - [ /3 width=10 by cpr_conf_lpr_bind_bind/ - | /4 width=11 by ex2_commute, cpr_conf_lpr_bind_zeta/ - | /3 width=11 by cpr_conf_lpr_bind_zeta/ - | /3 width=12 by cpr_conf_lpr_zeta_zeta/ - ] -| #I #V0 #T0 #Hn #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct - elim (cpr_inv_flat1 … H1) -H1 * - [ #V1 #T1 #HV01 #HT01 #H1 - | #HX1 #H1 - | #a1 #V1 #Y1 #Z1 #T1 #HV01 #HZT1 #H11 #H12 #H13 - | #a1 #V1 #U1 #Y1 #W1 #Z1 #T1 #HV01 #HVU1 #HYW1 #HZT1 #H11 #H12 #H13 - ] - elim (cpr_inv_flat1 … H2) -H2 * - [1,5,9,13: #V2 #T2 #HV02 #HT02 #H2 - |2,6,10,14: #HX2 #H2 - |3,7,11,15: #a2 #V2 #Y2 #Z2 #T2 #HV02 #HZT2 #H21 #H22 #H23 - |4,8,12,16: #a2 #V2 #U2 #Y2 #W2 #Z2 #T2 #HV02 #HVU2 #HYW2 #HZT2 #H21 #H22 #H23 - ] destruct - [ /3 width=10 by cpr_conf_lpr_flat_flat/ - | /4 width=8 by ex2_commute, cpr_conf_lpr_flat_tau/ - | /4 width=11 by ex2_commute, cpr_conf_lpr_flat_beta/ - | /4 width=14 by ex2_commute, cpr_conf_lpr_flat_theta/ - | /3 width=8 by cpr_conf_lpr_flat_tau/ - | /3 width=7 by cpr_conf_lpr_tau_tau/ - | /3 width=11 by cpr_conf_lpr_flat_beta/ - | /3 width=11 by cpr_conf_lpr_beta_beta/ - | /3 width=14 by cpr_conf_lpr_flat_theta/ - | /3 width=17 by cpr_conf_lpr_theta_theta/ - ] -] -qed-. - -(* Basic_1: includes: pr0_confluence pr2_confluence *) -theorem cpr_conf: ∀L. confluent … (cpr L). -/2 width=6 by cpr_conf_lpr/ qed-. - -(* Properties on context-sensitive parallel reduction for terms *************) - -lemma lpr_cpr_conf_dx: ∀L0,T0,T1. L0 ⊢ T0 ➡ T1 → ∀L1. L0 ⊢ ➡ L1 → - ∃∃T. L1 ⊢ T0 ➡ T & L1 ⊢ T1 ➡ T. -#L0 #T0 #T1 #HT01 #L1 #HL01 -elim (cpr_conf_lpr … HT01 T0 … HL01 … HL01) // -L0 /2 width=3/ -qed-. - -lemma lpr_cpr_conf_sn: ∀L0,T0,T1. L0 ⊢ T0 ➡ T1 → ∀L1. L0 ⊢ ➡ L1 → - ∃∃T. L1 ⊢ T0 ➡ T & L0 ⊢ T1 ➡ T. -#L0 #T0 #T1 #HT01 #L1 #HL01 -elim (cpr_conf_lpr … HT01 T0 … L0 … HL01) // -HT01 -HL01 /2 width=3/ -qed-. - -lemma fsup_cpr_trans: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ → ∀U2. L2 ⊢ T2 ➡ U2 → - ∃∃L,U1. L1 ⊢ ➡ L & L ⊢ T1 ➡ U1 & ⦃L, U1⦄ ⊃ ⦃L2, U2⦄. -#L1 #L2 #T1 #T2 #H elim H -L1 -L2 -T1 -T2 [1,2,3,4,5: /3 width=5/ ] -#L1 #K1 #K2 #T1 #T2 #U1 #d #e #HLK1 #HTU1 #_ #IHT12 #U2 #HTU2 -elim (IHT12 … HTU2) -IHT12 -HTU2 #K #T #HK1 #HT1 #HT2 -elim (lift_total T d e) #U #HTU -elim (ldrop_lpr_trans … HLK1 … HK1) -HLK1 -HK1 #L2 #HL12 #HL2K -lapply (cpr_lift … HT1 … HL2K … HTU1 … HTU) -HT1 -HTU1 /3 width=11/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/lpr_cpss.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/lpr_cpss.ma deleted file mode 100644 index 955108572..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/reduction/lpr_cpss.ma +++ /dev/null @@ -1,271 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/grammar/lpx_sn_lpx_sn.ma". -include "basic_2/substitution/lpss_ldrop.ma". -include "basic_2/reduction/lpr_ldrop.ma". - -(* SN PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS *****************************) - -(* Properties on context-sensitive parallel substitution for terms **********) - -fact cpr_cpss_conf_lpr_lpss_atom_atom: - ∀I,L1,L2. ∃∃T. L1 ⊢ ⓪{I} ▶* T & L2 ⊢ ⓪{I} ➡ T. -/2 width=3/ qed-. - -fact cpr_cpss_conf_lpr_lpss_atom_delta: - ∀L0,i. ( - ∀L,T.♯{L, T} < ♯{L0, #i} → - ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 → - ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 → - ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0 - ) → - ∀K0,V0. ⇩[O, i] L0 ≡ K0.ⓓV0 → - ∀V2. K0 ⊢ V0 ▶* V2 → ∀T2. ⇧[O, i + 1] V2 ≡ T2 → - ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 → - ∃∃T. L1 ⊢ #i ▶* T & L2 ⊢ T2 ➡ T. -#L0 #i #IH #K0 #V0 #HLK0 #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02 -elim (lpr_ldrop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1 -elim (lpr_inv_pair1 … H1) -H1 #K1 #V1 #HK01 #HV01 #H destruct -elim (lpss_ldrop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2 -elim (lpss_inv_pair1 … H2) -H2 #K2 #W2 #HK02 #_ #H destruct -lapply (ldrop_fwd_ldrop2 … HLK2) -W2 #HLK2 -lapply (ldrop_pair2_fwd_fw … HLK0 (#i)) -HLK0 #HLK0 -elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2 -elim (lift_total V 0 (i+1)) #T #HVT -lapply (cpr_lift … HV2 … HLK2 … HVT2 … HVT) -K2 -V2 /3 width=6/ -qed-. - -fact cpr_cpss_conf_lpr_lpss_delta_atom: - ∀L0,i. ( - ∀L,T.♯{L, T} < ♯{L0, #i} → - ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 → - ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 → - ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0 - ) → - ∀K0,V0. ⇩[O, i] L0 ≡ K0.ⓓV0 → - ∀V1. K0 ⊢ V0 ➡ V1 → ∀T1. ⇧[O, i + 1] V1 ≡ T1 → - ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 → - ∃∃T. L1 ⊢ T1 ▶* T & L2 ⊢ #i ➡ T. -#L0 #i #IH #K0 #V0 #HLK0 #V1 #HV01 #T1 #HVT1 #L1 #HL01 #L2 #HL02 -elim (lpss_ldrop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2 -elim (lpss_inv_pair1 … H2) -H2 #K2 #V2 #HK02 #HV02 #H destruct -elim (lpr_ldrop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1 -elim (lpr_inv_pair1 … H1) -H1 #K1 #W1 #HK01 #_ #H destruct -lapply (ldrop_fwd_ldrop2 … HLK1) -W1 #HLK1 -lapply (ldrop_pair2_fwd_fw … HLK0 (#i)) -HLK0 #HLK0 -elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2 -elim (lift_total V 0 (i+1)) #T #HVT -lapply (cpss_lift … HV1 … HLK1 … HVT1 … HVT) -K1 -V1 /3 width=9/ -qed-. - -fact cpr_cpss_conf_lpr_lpss_delta_delta: - ∀L0,i. ( - ∀L,T.♯{L, T} < ♯{L0, #i} → - ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 → - ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 → - ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0 - ) → - ∀K0,V0. ⇩[O, i] L0 ≡ K0.ⓓV0 → - ∀V1. K0 ⊢ V0 ➡ V1 → ∀T1. ⇧[O, i + 1] V1 ≡ T1 → - ∀KX,VX. ⇩[O, i] L0 ≡ KX.ⓓVX → - ∀V2. KX ⊢ VX ▶* V2 → ∀T2. ⇧[O, i + 1] V2 ≡ T2 → - ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 → - ∃∃T. L1 ⊢ T1 ▶* T & L2 ⊢ T2 ➡ T. -#L0 #i #IH #K0 #V0 #HLK0 #V1 #HV01 #T1 #HVT1 -#KX #VX #H #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02 -lapply (ldrop_mono … H … HLK0) -H #H destruct -elim (lpr_ldrop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1 -elim (lpr_inv_pair1 … H1) -H1 #K1 #W1 #HK01 #_ #H destruct -lapply (ldrop_fwd_ldrop2 … HLK1) -W1 #HLK1 -elim (lpss_ldrop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2 -elim (lpss_inv_pair1 … H2) -H2 #K2 #W2 #HK02 #_ #H destruct -lapply (ldrop_fwd_ldrop2 … HLK2) -W2 #HLK2 -lapply (ldrop_pair2_fwd_fw … HLK0 (#i)) -HLK0 #HLK0 -elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2 -elim (lift_total V 0 (i+1)) #T #HVT -lapply (cpss_lift … HV1 … HLK1 … HVT1 … HVT) -K1 -V1 -lapply (cpr_lift … HV2 … HLK2 … HVT2 … HVT) -K2 -V2 -V /2 width=3/ -qed-. - -fact cpr_cpss_conf_lpr_lpss_bind_bind: - ∀a,I,L0,V0,T0. ( - ∀L,T.♯{L,T} < ♯{L0,ⓑ{a,I}V0.T0} → - ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 → - ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 → - ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0 - ) → - ∀V1. L0 ⊢ V0 ➡ V1 → ∀T1. L0.ⓑ{I}V0 ⊢ T0 ➡ T1 → - ∀V2. L0 ⊢ V0 ▶* V2 → ∀T2. L0.ⓑ{I}V0 ⊢ T0 ▶* T2 → - ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 → - ∃∃T. L1 ⊢ ⓑ{a,I}V1.T1 ▶* T & L2 ⊢ ⓑ{a,I}V2.T2 ➡ T. -#a #I #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01 -#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02 -elim (IH … HV01 … HV02 … HL01 … HL02) // -elim (IH … HT01 … HT02 (L1.ⓑ{I}V1) … (L2.ⓑ{I}V2)) -IH // /2 width=1/ /3 width=5/ -qed-. - -fact cpr_cpss_conf_lpr_lpss_bind_zeta: - ∀L0,V0,T0. ( - ∀L,T.♯{L,T} < ♯{L0,+ⓓV0.T0} → - ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 → - ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 → - ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0 - ) → - ∀T1. L0.ⓓV0 ⊢ T0 ➡ T1 → ∀X1. ⇧[O, 1] X1 ≡ T1 → - ∀V2. L0 ⊢ V0 ▶* V2 → ∀T2. L0.ⓓV0 ⊢ T0 ▶* T2 → - ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 → - ∃∃T. L1 ⊢ X1 ▶* T & L2 ⊢ +ⓓV2.T2 ➡ T. -#L0 #V0 #T0 #IH #T1 #HT01 #X1 #HXT1 -#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02 -elim (IH … HT01 … HT02 (L1.ⓓV2) … (L2.ⓓV2)) -IH -HT01 -HT02 // /2 width=1/ /3 width=1/ -L0 -V0 -T0 #T #HT1 #HT2 -elim (cpss_inv_lift1 … HT1 L1 … HXT1) -T1 /2 width=1/ /3 width=9/ -qed-. - -fact cpr_cpss_conf_lpr_lpss_flat_flat: - ∀I,L0,V0,T0. ( - ∀L,T.♯{L,T} < ♯{L0,ⓕ{I}V0.T0} → - ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 → - ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 → - ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0 - ) → - ∀V1. L0 ⊢ V0 ➡ V1 → ∀T1. L0 ⊢ T0 ➡ T1 → - ∀V2. L0 ⊢ V0 ▶* V2 → ∀T2. L0 ⊢ T0 ▶* T2 → - ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 → - ∃∃T. L1 ⊢ ⓕ{I}V1.T1 ▶* T & L2 ⊢ ⓕ{I}V2.T2 ➡ T. -#I #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01 -#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02 -elim (IH … HV01 … HV02 … HL01 … HL02) // -elim (IH … HT01 … HT02 … HL01 … HL02) // /3 width=5/ -qed-. - -fact cpr_cpss_conf_lpr_lpss_flat_tau: - ∀L0,V0,T0. ( - ∀L,T.♯{L,T} < ♯{L0,ⓝV0.T0} → - ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 → - ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 → - ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0 - ) → - ∀T1. L0 ⊢ T0 ➡ T1 → ∀V2,T2. L0 ⊢ T0 ▶* T2 → - ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 → - ∃∃T. L1 ⊢ T1 ▶* T & L2 ⊢ ⓝV2.T2 ➡ T. -#L0 #V0 #T0 #IH #T1 #HT01 -#V2 #T2 #HT02 #L1 #HL01 #L2 #HL02 -elim (IH … HT01 … HT02 … HL01 … HL02) // -L0 -V0 -T0 /3 width=3/ -qed-. - -fact cpr_cpss_conf_lpr_lpss_flat_beta: - ∀a,L0,V0,W0,T0. ( - ∀L,T.♯{L,T} < ♯{L0,ⓐV0.ⓛ{a}W0.T0} → - ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 → - ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 → - ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0 - ) → - ∀V1. L0 ⊢ V0 ➡ V1 → ∀T1. L0.ⓛW0 ⊢ T0 ➡ T1 → - ∀V2. L0 ⊢ V0 ▶* V2 → ∀T2. L0 ⊢ ⓛ{a}W0.T0 ▶* T2 → - ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 → - ∃∃T. L1 ⊢ ⓓ{a}V1.T1 ▶* T & L2 ⊢ ⓐV2.T2 ➡ T. -#a #L0 #V0 #W0 #T0 #IH #V1 #HV01 #T1 #HT01 -#V2 #HV02 #X #H #L1 #HL01 #L2 #HL02 -elim (cpss_inv_bind1 … H) -H #W2 #T2 #HW02 #HT02 #H destruct -elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1/ #V #HV1 #HV2 -elim (IH … HT01 … HT02 (L1.ⓛW2) … (L2.ⓛW2)) /2 width=1/ /3 width=1/ -L0 -V0 -W0 -T0 #T #HT1 #HT2 -lapply (cpss_lsubr_trans … HT1 (L1.ⓓV1) ?) -HT1 /2 width=1/ /3 width=5/ -qed-. - -fact cpr_cpss_conf_lpr_lpss_flat_theta: - ∀a,L0,V0,W0,T0. ( - ∀L,T.♯{L,T} < ♯{L0,ⓐV0.ⓓ{a}W0.T0} → - ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 → - ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 → - ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0 - ) → - ∀V1. L0 ⊢ V0 ➡ V1 → ∀U1. ⇧[O, 1] V1 ≡ U1 → - ∀W1. L0 ⊢ W0 ➡ W1 → ∀T1. L0.ⓓW0 ⊢ T0 ➡ T1 → - ∀V2. L0 ⊢ V0 ▶* V2 → ∀T2. L0 ⊢ ⓓ{a}W0.T0 ▶* T2 → - ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 → - ∃∃T. L1 ⊢ ⓓ{a}W1.ⓐU1.T1 ▶* T & L2 ⊢ ⓐV2.T2 ➡ T. -#a #L0 #V0 #W0 #T0 #IH #V1 #HV01 #U1 #HVU1 #W1 #HW01 #T1 #HT01 -#V2 #HV02 #X #H #L1 #HL01 #L2 #HL02 -elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1/ #V #HV1 #HV2 -elim (lift_total V 0 1) #U #HVU -lapply (cpss_lift … HV1 (L1.ⓓW1) … HVU1 … HVU) -HVU1 /2 width=1/ #HU1 -elim (cpss_inv_bind1 … H) -H #W2 #T2 #HW02 #HT02 #H destruct -elim (IH … HW01 … HW02 … HL01 … HL02) /2 width=1/ -elim (IH … HT01 … HT02 (L1.ⓓW1) … (L2.ⓓW2)) /2 width=1/ -L0 -V0 -W0 -T0 -/4 width=9 by ex2_intro, cpr_theta, cpss_bind, cpss_flat/ (**) (* auto too slow without trace *) -qed-. - -lemma cpr_cpss_conf_lpr_lpss: lpx_sn_confluent cpr cpss. -#L0 #T0 @(f2_ind … fw … L0 T0) -L0 -T0 #n #IH #L0 * [|*] -[ #I0 #Hn #T1 #H1 #T2 #H2 #L1 #HL01 #L2 #HL02 destruct - elim (cpr_inv_atom1 … H1) -H1 - elim (cpss_inv_atom1 … H2) -H2 - [ #H2 #H1 destruct - /2 width=1 by cpr_cpss_conf_lpr_lpss_atom_atom/ - | * #K0 #V0 #V2 #i2 #HLK0 #HV02 #HVT2 #H2 #H1 destruct - /3 width=10 by cpr_cpss_conf_lpr_lpss_atom_delta/ - | #H2 * #K0 #V0 #V1 #i1 #HLK0 #HV01 #HVT1 #H1 destruct - /3 width=10 by cpr_cpss_conf_lpr_lpss_delta_atom/ - | * #X #Y #V2 #z #H #HV02 #HVT2 #H2 - * #K0 #V0 #V1 #i #HLK0 #HV01 #HVT1 #H1 destruct - /3 width=17 by cpr_cpss_conf_lpr_lpss_delta_delta/ - ] -| #a #I #V0 #T0 #Hn #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct - elim (cpss_inv_bind1 … H2) -H2 #V2 #T2 #HV02 #HT02 #H2 - elim (cpr_inv_bind1 … H1) -H1 * - [ #V1 #T1 #HV01 #HT01 #H1 - | #T1 #HT01 #HXT1 #H11 #H12 - ] destruct - [ /3 width=10 by cpr_cpss_conf_lpr_lpss_bind_bind/ - | /3 width=11 by cpr_cpss_conf_lpr_lpss_bind_zeta/ - ] -| #I #V0 #T0 #Hn #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct - elim (cpss_inv_flat1 … H2) -H2 #V2 #T2 #HV02 #HT02 #H2 - elim (cpr_inv_flat1 … H1) -H1 * - [ #V1 #T1 #HV01 #HT01 #H1 - | #HX1 #H1 - | #a1 #V1 #Y1 #Z1 #T1 #HV01 #HZT1 #H11 #H12 #H13 - | #a1 #V1 #U1 #Y1 #W1 #Z1 #T1 #HV01 #HVU1 #HYW1 #HZT1 #H11 #H12 #H13 - ] destruct - [ /3 width=10 by cpr_cpss_conf_lpr_lpss_flat_flat/ - | /3 width=8 by cpr_cpss_conf_lpr_lpss_flat_tau/ - | /3 width=11 by cpr_cpss_conf_lpr_lpss_flat_beta/ - | /3 width=14 by cpr_cpss_conf_lpr_lpss_flat_theta/ - ] -] -qed-. - -(* Basic_1: includes: pr0_subst1 *) -lemma cpr_cpss_conf: ∀L. confluent2 … (cpr L) (cpss L). -/2 width=6 by cpr_cpss_conf_lpr_lpss/ qed-. - -lemma cpr_lpss_conf_dx: ∀L0,T0,T1. L0 ⊢ T0 ➡ T1 → ∀L1. L0 ⊢ ▶* L1 → - ∃∃T. L1 ⊢ T1 ▶* T & L1 ⊢ T0 ➡ T. -#L0 #T0 #T1 #HT01 #L1 #HL01 -elim (cpr_cpss_conf_lpr_lpss … HT01 T0 … L1 … HL01) // /2 width=1/ -L0 /2 width=3/ -qed-. - -lemma cpr_lpss_conf_sn: ∀L0,T0,T1. L0 ⊢ T0 ➡ T1 → ∀L1. L0 ⊢ ▶* L1 → - ∃∃T. L0 ⊢ T1 ▶* T & L1 ⊢ T0 ➡ T. -#L0 #T0 #T1 #HT01 #L1 #HL01 -elim (cpr_cpss_conf_lpr_lpss … HT01 T0 … L0 … HL01) // -HT01 -HL01 /2 width=3/ -qed-. - -(* Basic_1: includes: pr0_subst1_fwd *) -lemma lpr_cpss_conf: ∀L0,T0,T1. L0 ⊢ T0 ▶* T1 → ∀L1. L0 ⊢ ➡ L1 → - ∃∃T. L1 ⊢ T0 ▶* T & L0 ⊢ T1 ➡ T. -#L0 #T0 #T1 #HT01 #L1 #HL01 -elim (cpr_cpss_conf_lpr_lpss ?? T0 … HT01 … HL01 L0) // -HT01 -HL01 /2 width=3/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/lpr_ldrop.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/lpr_ldrop.ma index 2ea40eeb4..412d42207 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/reduction/lpr_ldrop.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/lpr_ldrop.ma @@ -12,6 +12,7 @@ (* *) (**************************************************************************) +include "basic_2/relocation/fsup.ma". include "basic_2/relocation/ldrop_lpx_sn.ma". include "basic_2/reduction/cpr_lift.ma". include "basic_2/reduction/lpr.ma". @@ -30,3 +31,15 @@ lemma ldrop_lpr_trans: dedropable_sn lpr. lemma lpr_ldrop_trans_O1: dropable_dx lpr. /2 width=3 by lpx_sn_dropable/ qed-. + +(* Properties on context-sensitive parallel reduction for terms *************) + +lemma fsup_cpr_trans: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ → ∀U2. L2 ⊢ T2 ➡ U2 → + ∃∃L,U1. L1 ⊢ ➡ L & L ⊢ T1 ➡ U1 & ⦃L, U1⦄ ⊃ ⦃L2, U2⦄. +#L1 #L2 #T1 #T2 #H elim H -L1 -L2 -T1 -T2 [1,2,3,4,5: /3 width=5/ ] +#L1 #K1 #K2 #T1 #T2 #U1 #d #e #HLK1 #HTU1 #_ #IHT12 #U2 #HTU2 +elim (IHT12 … HTU2) -IHT12 -HTU2 #K #T #HK1 #HT1 #HT2 +elim (lift_total T d e) #U #HTU +elim (ldrop_lpr_trans … HLK1 … HK1) -HLK1 -HK1 #L2 #HL12 #HL2K +lapply (cpr_lift … HT1 … HL2K … HTU1 … HTU) -HT1 -HTU1 /3 width=11/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/lpr_lpr.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/lpr_lpr.ma index ab77692dd..d11e148d1 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/reduction/lpr_lpr.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/lpr_lpr.ma @@ -12,9 +12,338 @@ (* *) (**************************************************************************) -include "basic_2/reduction/lpr_cpr.ma". +include "basic_2/grammar/lpx_sn_lpx_sn.ma". +include "basic_2/relocation/fsup.ma". +include "basic_2/reduction/lpr_ldrop.ma". -(* SN PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ******************************) +(* SN PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS *****************************) + +(* Main properties on context-sensitive parallel reduction for terms ********) + +fact cpr_conf_lpr_atom_atom: + ∀I,L1,L2. ∃∃T. L1 ⊢ ⓪{I} ➡ T & L2 ⊢ ⓪{I} ➡ T. +/2 width=3/ qed-. + +fact cpr_conf_lpr_atom_delta: + ∀L0,i. ( + ∀L,T.♯{L, T} < ♯{L0, #i} → + ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 → + ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 → + ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0 + ) → + ∀K0,V0. ⇩[O, i] L0 ≡ K0.ⓓV0 → + ∀V2. K0 ⊢ V0 ➡ V2 → ∀T2. ⇧[O, i + 1] V2 ≡ T2 → + ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ➡ L2 → + ∃∃T. L1 ⊢ #i ➡ T & L2 ⊢ T2 ➡ T. +#L0 #i #IH #K0 #V0 #HLK0 #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02 +elim (lpr_ldrop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1 +elim (lpr_inv_pair1 … H1) -H1 #K1 #V1 #HK01 #HV01 #H destruct +elim (lpr_ldrop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2 +elim (lpr_inv_pair1 … H2) -H2 #K2 #W2 #HK02 #_ #H destruct +lapply (ldrop_fwd_ldrop2 … HLK2) -W2 #HLK2 +lapply (ldrop_pair2_fwd_fw … HLK0 (#i)) -HLK0 #HLK0 +elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2 +elim (lift_total V 0 (i+1)) #T #HVT +lapply (cpr_lift … HV2 … HLK2 … HVT2 … HVT) -K2 -V2 /3 width=6/ +qed-. + +(* Basic_1: includes: pr0_delta_delta pr2_delta_delta *) +fact cpr_conf_lpr_delta_delta: + ∀L0,i. ( + ∀L,T.♯{L, T} < ♯{L0, #i} → + ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 → + ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 → + ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0 + ) → + ∀K0,V0. ⇩[O, i] L0 ≡ K0.ⓓV0 → + ∀V1. K0 ⊢ V0 ➡ V1 → ∀T1. ⇧[O, i + 1] V1 ≡ T1 → + ∀KX,VX. ⇩[O, i] L0 ≡ KX.ⓓVX → + ∀V2. KX ⊢ VX ➡ V2 → ∀T2. ⇧[O, i + 1] V2 ≡ T2 → + ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ➡ L2 → + ∃∃T. L1 ⊢ T1 ➡ T & L2 ⊢ T2 ➡ T. +#L0 #i #IH #K0 #V0 #HLK0 #V1 #HV01 #T1 #HVT1 +#KX #VX #H #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02 +lapply (ldrop_mono … H … HLK0) -H #H destruct +elim (lpr_ldrop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1 +elim (lpr_inv_pair1 … H1) -H1 #K1 #W1 #HK01 #_ #H destruct +lapply (ldrop_fwd_ldrop2 … HLK1) -W1 #HLK1 +elim (lpr_ldrop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2 +elim (lpr_inv_pair1 … H2) -H2 #K2 #W2 #HK02 #_ #H destruct +lapply (ldrop_fwd_ldrop2 … HLK2) -W2 #HLK2 +lapply (ldrop_pair2_fwd_fw … HLK0 (#i)) -HLK0 #HLK0 +elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2 +elim (lift_total V 0 (i+1)) #T #HVT +lapply (cpr_lift … HV1 … HLK1 … HVT1 … HVT) -K1 -V1 +lapply (cpr_lift … HV2 … HLK2 … HVT2 … HVT) -K2 -V2 -V /2 width=3/ +qed-. + +fact cpr_conf_lpr_bind_bind: + ∀a,I,L0,V0,T0. ( + ∀L,T.♯{L,T} < ♯{L0,ⓑ{a,I}V0.T0} → + ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 → + ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 → + ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0 + ) → + ∀V1. L0 ⊢ V0 ➡ V1 → ∀T1. L0.ⓑ{I}V0 ⊢ T0 ➡ T1 → + ∀V2. L0 ⊢ V0 ➡ V2 → ∀T2. L0.ⓑ{I}V0 ⊢ T0 ➡ T2 → + ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ➡ L2 → + ∃∃T. L1 ⊢ ⓑ{a,I}V1.T1 ➡ T & L2 ⊢ ⓑ{a,I}V2.T2 ➡ T. +#a #I #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01 +#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02 +elim (IH … HV01 … HV02 … HL01 … HL02) // +elim (IH … HT01 … HT02 (L1.ⓑ{I}V1) … (L2.ⓑ{I}V2)) -IH // /2 width=1/ /3 width=5/ +qed-. + +fact cpr_conf_lpr_bind_zeta: + ∀L0,V0,T0. ( + ∀L,T.♯{L,T} < ♯{L0,+ⓓV0.T0} → + ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 → + ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 → + ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0 + ) → + ∀V1. L0 ⊢ V0 ➡ V1 → ∀T1. L0.ⓓV0 ⊢ T0 ➡ T1 → + ∀T2. L0.ⓓV0 ⊢ T0 ➡ T2 → ∀X2. ⇧[O, 1] X2 ≡ T2 → + ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ➡ L2 → + ∃∃T. L1 ⊢ +ⓓV1.T1 ➡ T & L2 ⊢ X2 ➡ T. +#L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01 +#T2 #HT02 #X2 #HXT2 #L1 #HL01 #L2 #HL02 +elim (IH … HT01 … HT02 (L1.ⓓV1) … (L2.ⓓV1)) -IH -HT01 -HT02 // /2 width=1/ -L0 -V0 -T0 #T #HT1 #HT2 +elim (cpr_inv_lift1 … HT2 L2 … HXT2) -T2 /2 width=1/ /3 width=3/ +qed-. + +fact cpr_conf_lpr_zeta_zeta: + ∀L0,V0,T0. ( + ∀L,T.♯{L,T} < ♯{L0,+ⓓV0.T0} → + ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 → + ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 → + ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0 + ) → + ∀T1. L0.ⓓV0 ⊢ T0 ➡ T1 → ∀X1. ⇧[O, 1] X1 ≡ T1 → + ∀T2. L0.ⓓV0 ⊢ T0 ➡ T2 → ∀X2. ⇧[O, 1] X2 ≡ T2 → + ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ➡ L2 → + ∃∃T. L1 ⊢ X1 ➡ T & L2 ⊢ X2 ➡ T. +#L0 #V0 #T0 #IH #T1 #HT01 #X1 #HXT1 +#T2 #HT02 #X2 #HXT2 #L1 #HL01 #L2 #HL02 +elim (IH … HT01 … HT02 (L1.ⓓV0) … (L2.ⓓV0)) -IH -HT01 -HT02 // /2 width=1/ -L0 -T0 #T #HT1 #HT2 +elim (cpr_inv_lift1 … HT1 L1 … HXT1) -T1 /2 width=1/ #T1 #HT1 #HXT1 +elim (cpr_inv_lift1 … HT2 L2 … HXT2) -T2 /2 width=1/ #T2 #HT2 #HXT2 +lapply (lift_inj … HT2 … HT1) -T #H destruct /2 width=3/ +qed-. + +fact cpr_conf_lpr_flat_flat: + ∀I,L0,V0,T0. ( + ∀L,T.♯{L,T} < ♯{L0,ⓕ{I}V0.T0} → + ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 → + ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 → + ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0 + ) → + ∀V1. L0 ⊢ V0 ➡ V1 → ∀T1. L0 ⊢ T0 ➡ T1 → + ∀V2. L0 ⊢ V0 ➡ V2 → ∀T2. L0 ⊢ T0 ➡ T2 → + ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ➡ L2 → + ∃∃T. L1 ⊢ ⓕ{I}V1.T1 ➡ T & L2 ⊢ ⓕ{I}V2.T2 ➡ T. +#I #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01 +#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02 +elim (IH … HV01 … HV02 … HL01 … HL02) // +elim (IH … HT01 … HT02 … HL01 … HL02) // /3 width=5/ +qed-. + +fact cpr_conf_lpr_flat_tau: + ∀L0,V0,T0. ( + ∀L,T.♯{L,T} < ♯{L0,ⓝV0.T0} → + ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 → + ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 → + ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0 + ) → + ∀V1,T1. L0 ⊢ T0 ➡ T1 → ∀T2. L0 ⊢ T0 ➡ T2 → + ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ➡ L2 → + ∃∃T. L1 ⊢ ⓝV1.T1 ➡ T & L2 ⊢ T2 ➡ T. +#L0 #V0 #T0 #IH #V1 #T1 #HT01 +#T2 #HT02 #L1 #HL01 #L2 #HL02 +elim (IH … HT01 … HT02 … HL01 … HL02) // -L0 -V0 -T0 /3 width=3/ +qed-. + +fact cpr_conf_lpr_tau_tau: + ∀L0,V0,T0. ( + ∀L,T.♯{L,T} < ♯{L0,ⓝV0.T0} → + ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 → + ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 → + ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0 + ) → + ∀T1. L0 ⊢ T0 ➡ T1 → ∀T2. L0 ⊢ T0 ➡ T2 → + ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ➡ L2 → + ∃∃T. L1 ⊢ T1 ➡ T & L2 ⊢ T2 ➡ T. +#L0 #V0 #T0 #IH #T1 #HT01 +#T2 #HT02 #L1 #HL01 #L2 #HL02 +elim (IH … HT01 … HT02 … HL01 … HL02) // -L0 -V0 -T0 /2 width=3/ +qed-. + +fact cpr_conf_lpr_flat_beta: + ∀a,L0,V0,W0,T0. ( + ∀L,T.♯{L,T} < ♯{L0,ⓐV0.ⓛ{a}W0.T0} → + ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 → + ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 → + ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0 + ) → + ∀V1. L0 ⊢ V0 ➡ V1 → ∀T1. L0 ⊢ ⓛ{a}W0.T0 ➡ T1 → + ∀V2. L0 ⊢ V0 ➡ V2 → ∀T2. L0.ⓛW0 ⊢ T0 ➡ T2 → + ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ➡ L2 → + ∃∃T. L1 ⊢ ⓐV1.T1 ➡ T & L2 ⊢ ⓓ{a}V2.T2 ➡ T. +#a #L0 #V0 #W0 #T0 #IH #V1 #HV01 #X #H +#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02 +elim (cpr_inv_abst1 … H) -H #W1 #T1 #HW01 #HT01 #H destruct +elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1/ #V #HV1 #HV2 +elim (IH … HT01 … HT02 (L1.ⓛW1) … (L2.ⓛW1)) /2 width=1/ -L0 -V0 -W0 -T0 #T #HT1 #HT2 +lapply (cpr_lsubr_trans … HT2 (L2.ⓓV2) ?) -HT2 /2 width=1/ /3 width=5/ +qed-. + +(* Basic-1: includes: + pr0_cong_upsilon_refl pr0_cong_upsilon_zeta + pr0_cong_upsilon_cong pr0_cong_upsilon_delta +*) +fact cpr_conf_lpr_flat_theta: + ∀a,L0,V0,W0,T0. ( + ∀L,T.♯{L,T} < ♯{L0,ⓐV0.ⓓ{a}W0.T0} → + ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 → + ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 → + ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0 + ) → + ∀V1. L0 ⊢ V0 ➡ V1 → ∀T1. L0 ⊢ ⓓ{a}W0.T0 ➡ T1 → + ∀V2. L0 ⊢ V0 ➡ V2 → ∀U2. ⇧[O, 1] V2 ≡ U2 → + ∀W2. L0 ⊢ W0 ➡ W2 → ∀T2. L0.ⓓW0 ⊢ T0 ➡ T2 → + ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ➡ L2 → + ∃∃T. L1 ⊢ ⓐV1.T1 ➡ T & L2 ⊢ ⓓ{a}W2.ⓐU2.T2 ➡ T. +#a #L0 #V0 #W0 #T0 #IH #V1 #HV01 #X #H +#V2 #HV02 #U2 #HVU2 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02 +elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1/ #V #HV1 #HV2 +elim (lift_total V 0 1) #U #HVU +lapply (cpr_lift … HV2 (L2.ⓓW2) … HVU2 … HVU) -HVU2 /2 width=1/ #HU2 +elim (cpr_inv_abbr1 … H) -H * +[ #W1 #T1 #HW01 #HT01 #H destruct + elim (IH … HW01 … HW02 … HL01 … HL02) /2 width=1/ + elim (IH … HT01 … HT02 (L1.ⓓW1) … (L2.ⓓW2)) /2 width=1/ -L0 -V0 -W0 -T0 /4 width=7/ +| #T1 #HT01 #HXT1 #H destruct + elim (IH … HT01 … HT02 (L1.ⓓW2) … (L2.ⓓW2)) /2 width=1/ -L0 -V0 -W0 -T0 #T #HT1 #HT2 + elim (cpr_inv_lift1 … HT1 L1 … HXT1) -HXT1 /2 width=1/ #Y #HYT #HXY + @(ex2_intro … (ⓐV.Y)) /2 width=1/ /3 width=5/ (**) (* auto /4 width=9/ is too slow *) +] +qed-. + +fact cpr_conf_lpr_beta_beta: + ∀a,L0,V0,W0,T0. ( + ∀L,T.♯{L,T} < ♯{L0,ⓐV0.ⓛ{a}W0.T0} → + ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 → + ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 → + ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0 + ) → + ∀V1. L0 ⊢ V0 ➡ V1 → ∀T1. L0.ⓛW0 ⊢ T0 ➡ T1 → + ∀V2. L0 ⊢ V0 ➡ V2 → ∀T2. L0.ⓛW0 ⊢ T0 ➡ T2 → + ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ➡ L2 → + ∃∃T. L1 ⊢ ⓓ{a}V1.T1 ➡ T & L2 ⊢ ⓓ{a}V2.T2 ➡ T. +#a #L0 #V0 #W0 #T0 #IH #V1 #HV01 #T1 #HT01 +#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02 +elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1/ #V #HV1 #HV2 +elim (IH … HT01 … HT02 (L1.ⓛW0) … (L2.ⓛW0)) /2 width=1/ -L0 -V0 -T0 #T #HT1 #HT2 +lapply (cpr_lsubr_trans … HT1 (L1.ⓓV1) ?) -HT1 /2 width=1/ +lapply (cpr_lsubr_trans … HT2 (L2.ⓓV2) ?) -HT2 /2 width=1/ /3 width=5/ +qed-. + +(* Basic_1: was: pr0_upsilon_upsilon *) +fact cpr_conf_lpr_theta_theta: + ∀a,L0,V0,W0,T0. ( + ∀L,T.♯{L,T} < ♯{L0,ⓐV0.ⓓ{a}W0.T0} → + ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ➡ T2 → + ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ➡ L2 → + ∃∃T0. L1 ⊢ T1 ➡ T0 & L2 ⊢ T2 ➡ T0 + ) → + ∀V1. L0 ⊢ V0 ➡ V1 → ∀U1. ⇧[O, 1] V1 ≡ U1 → + ∀W1. L0 ⊢ W0 ➡ W1 → ∀T1. L0.ⓓW0 ⊢ T0 ➡ T1 → + ∀V2. L0 ⊢ V0 ➡ V2 → ∀U2. ⇧[O, 1] V2 ≡ U2 → + ∀W2. L0 ⊢ W0 ➡ W2 → ∀T2. L0.ⓓW0 ⊢ T0 ➡ T2 → + ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ➡ L2 → + ∃∃T. L1 ⊢ ⓓ{a}W1.ⓐU1.T1 ➡ T & L2 ⊢ ⓓ{a}W2.ⓐU2.T2 ➡ T. +#a #L0 #V0 #W0 #T0 #IH #V1 #HV01 #U1 #HVU1 #W1 #HW01 #T1 #HT01 +#V2 #HV02 #U2 #HVU2 #W2 #HW02 #T2 #HT02 #L1 #HL01 #L2 #HL02 +elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1/ #V #HV1 #HV2 +elim (IH … HW01 … HW02 … HL01 … HL02) /2 width=1/ +elim (IH … HT01 … HT02 (L1.ⓓW1) … (L2.ⓓW2)) /2 width=1/ -L0 -V0 -W0 -T0 +elim (lift_total V 0 1) #U #HVU +lapply (cpr_lift … HV1 (L1.ⓓW1) … HVU1 … HVU) -HVU1 /2 width=1/ +lapply (cpr_lift … HV2 (L2.ⓓW2) … HVU2 … HVU) -HVU2 /2 width=1/ /4 width=7/ +qed-. + +theorem cpr_conf_lpr: lpx_sn_confluent cpr cpr. +#L0 #T0 @(f2_ind … fw … L0 T0) -L0 -T0 #n #IH #L0 * [|*] +[ #I0 #Hn #T1 #H1 #T2 #H2 #L1 #HL01 #L2 #HL02 destruct + elim (cpr_inv_atom1 … H1) -H1 + elim (cpr_inv_atom1 … H2) -H2 + [ #H2 #H1 destruct + /2 width=1 by cpr_conf_lpr_atom_atom/ + | * #K0 #V0 #V2 #i2 #HLK0 #HV02 #HVT2 #H2 #H1 destruct + /3 width=10 by cpr_conf_lpr_atom_delta/ + | #H2 * #K0 #V0 #V1 #i1 #HLK0 #HV01 #HVT1 #H1 destruct + /4 width=10 by ex2_commute, cpr_conf_lpr_atom_delta/ + | * #X #Y #V2 #z #H #HV02 #HVT2 #H2 + * #K0 #V0 #V1 #i #HLK0 #HV01 #HVT1 #H1 destruct + /3 width=17 by cpr_conf_lpr_delta_delta/ + ] +| #a #I #V0 #T0 #Hn #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct + elim (cpr_inv_bind1 … H1) -H1 * + [ #V1 #T1 #HV01 #HT01 #H1 + | #T1 #HT01 #HXT1 #H11 #H12 + ] + elim (cpr_inv_bind1 … H2) -H2 * + [1,3: #V2 #T2 #HV02 #HT02 #H2 + |2,4: #T2 #HT02 #HXT2 #H21 #H22 + ] destruct + [ /3 width=10 by cpr_conf_lpr_bind_bind/ + | /4 width=11 by ex2_commute, cpr_conf_lpr_bind_zeta/ + | /3 width=11 by cpr_conf_lpr_bind_zeta/ + | /3 width=12 by cpr_conf_lpr_zeta_zeta/ + ] +| #I #V0 #T0 #Hn #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct + elim (cpr_inv_flat1 … H1) -H1 * + [ #V1 #T1 #HV01 #HT01 #H1 + | #HX1 #H1 + | #a1 #V1 #Y1 #Z1 #T1 #HV01 #HZT1 #H11 #H12 #H13 + | #a1 #V1 #U1 #Y1 #W1 #Z1 #T1 #HV01 #HVU1 #HYW1 #HZT1 #H11 #H12 #H13 + ] + elim (cpr_inv_flat1 … H2) -H2 * + [1,5,9,13: #V2 #T2 #HV02 #HT02 #H2 + |2,6,10,14: #HX2 #H2 + |3,7,11,15: #a2 #V2 #Y2 #Z2 #T2 #HV02 #HZT2 #H21 #H22 #H23 + |4,8,12,16: #a2 #V2 #U2 #Y2 #W2 #Z2 #T2 #HV02 #HVU2 #HYW2 #HZT2 #H21 #H22 #H23 + ] destruct + [ /3 width=10 by cpr_conf_lpr_flat_flat/ + | /4 width=8 by ex2_commute, cpr_conf_lpr_flat_tau/ + | /4 width=11 by ex2_commute, cpr_conf_lpr_flat_beta/ + | /4 width=14 by ex2_commute, cpr_conf_lpr_flat_theta/ + | /3 width=8 by cpr_conf_lpr_flat_tau/ + | /3 width=7 by cpr_conf_lpr_tau_tau/ + | /3 width=11 by cpr_conf_lpr_flat_beta/ + | /3 width=11 by cpr_conf_lpr_beta_beta/ + | /3 width=14 by cpr_conf_lpr_flat_theta/ + | /3 width=17 by cpr_conf_lpr_theta_theta/ + ] +] +qed-. + +(* Basic_1: includes: pr0_confluence pr2_confluence *) +theorem cpr_conf: ∀L. confluent … (cpr L). +/2 width=6 by cpr_conf_lpr/ qed-. + +(* Properties on context-sensitive parallel reduction for terms *************) + +lemma lpr_cpr_conf_dx: ∀L0,T0,T1. L0 ⊢ T0 ➡ T1 → ∀L1. L0 ⊢ ➡ L1 → + ∃∃T. L1 ⊢ T0 ➡ T & L1 ⊢ T1 ➡ T. +#L0 #T0 #T1 #HT01 #L1 #HL01 +elim (cpr_conf_lpr … HT01 T0 … HL01 … HL01) // -L0 /2 width=3/ +qed-. + +lemma lpr_cpr_conf_sn: ∀L0,T0,T1. L0 ⊢ T0 ➡ T1 → ∀L1. L0 ⊢ ➡ L1 → + ∃∃T. L1 ⊢ T0 ➡ T & L0 ⊢ T1 ➡ T. +#L0 #T0 #T1 #HT01 #L1 #HL01 +elim (cpr_conf_lpr … HT01 T0 … L0 … HL01) // -HT01 -HL01 /2 width=3/ +qed-. (* Main properties **********************************************************) diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/lpr_lpss.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/lpr_lpss.ma index 34aaa4e12..6ea796b9a 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/reduction/lpr_lpss.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/lpr_lpss.ma @@ -12,9 +12,263 @@ (* *) (**************************************************************************) -include "basic_2/reduction/lpr_cpss.ma". +include "basic_2/grammar/lpx_sn_lpx_sn.ma". +include "basic_2/substitution/lpss_ldrop.ma". +include "basic_2/reduction/lpr_ldrop.ma". -(* SN PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ******************************) +(* SN PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS *****************************) + +(* Properties on context-sensitive parallel substitution for terms **********) + +fact cpr_cpss_conf_lpr_lpss_atom_atom: + ∀I,L1,L2. ∃∃T. L1 ⊢ ⓪{I} ▶* T & L2 ⊢ ⓪{I} ➡ T. +/2 width=3/ qed-. + +fact cpr_cpss_conf_lpr_lpss_atom_delta: + ∀L0,i. ( + ∀L,T.♯{L, T} < ♯{L0, #i} → + ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 → + ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 → + ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0 + ) → + ∀K0,V0. ⇩[O, i] L0 ≡ K0.ⓓV0 → + ∀V2. K0 ⊢ V0 ▶* V2 → ∀T2. ⇧[O, i + 1] V2 ≡ T2 → + ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 → + ∃∃T. L1 ⊢ #i ▶* T & L2 ⊢ T2 ➡ T. +#L0 #i #IH #K0 #V0 #HLK0 #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02 +elim (lpr_ldrop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1 +elim (lpr_inv_pair1 … H1) -H1 #K1 #V1 #HK01 #HV01 #H destruct +elim (lpss_ldrop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2 +elim (lpss_inv_pair1 … H2) -H2 #K2 #W2 #HK02 #_ #H destruct +lapply (ldrop_fwd_ldrop2 … HLK2) -W2 #HLK2 +lapply (ldrop_pair2_fwd_fw … HLK0 (#i)) -HLK0 #HLK0 +elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2 +elim (lift_total V 0 (i+1)) #T #HVT +lapply (cpr_lift … HV2 … HLK2 … HVT2 … HVT) -K2 -V2 /3 width=6/ +qed-. + +fact cpr_cpss_conf_lpr_lpss_delta_atom: + ∀L0,i. ( + ∀L,T.♯{L, T} < ♯{L0, #i} → + ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 → + ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 → + ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0 + ) → + ∀K0,V0. ⇩[O, i] L0 ≡ K0.ⓓV0 → + ∀V1. K0 ⊢ V0 ➡ V1 → ∀T1. ⇧[O, i + 1] V1 ≡ T1 → + ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 → + ∃∃T. L1 ⊢ T1 ▶* T & L2 ⊢ #i ➡ T. +#L0 #i #IH #K0 #V0 #HLK0 #V1 #HV01 #T1 #HVT1 #L1 #HL01 #L2 #HL02 +elim (lpss_ldrop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2 +elim (lpss_inv_pair1 … H2) -H2 #K2 #V2 #HK02 #HV02 #H destruct +elim (lpr_ldrop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1 +elim (lpr_inv_pair1 … H1) -H1 #K1 #W1 #HK01 #_ #H destruct +lapply (ldrop_fwd_ldrop2 … HLK1) -W1 #HLK1 +lapply (ldrop_pair2_fwd_fw … HLK0 (#i)) -HLK0 #HLK0 +elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2 +elim (lift_total V 0 (i+1)) #T #HVT +lapply (cpss_lift … HV1 … HLK1 … HVT1 … HVT) -K1 -V1 /3 width=9/ +qed-. + +fact cpr_cpss_conf_lpr_lpss_delta_delta: + ∀L0,i. ( + ∀L,T.♯{L, T} < ♯{L0, #i} → + ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 → + ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 → + ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0 + ) → + ∀K0,V0. ⇩[O, i] L0 ≡ K0.ⓓV0 → + ∀V1. K0 ⊢ V0 ➡ V1 → ∀T1. ⇧[O, i + 1] V1 ≡ T1 → + ∀KX,VX. ⇩[O, i] L0 ≡ KX.ⓓVX → + ∀V2. KX ⊢ VX ▶* V2 → ∀T2. ⇧[O, i + 1] V2 ≡ T2 → + ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 → + ∃∃T. L1 ⊢ T1 ▶* T & L2 ⊢ T2 ➡ T. +#L0 #i #IH #K0 #V0 #HLK0 #V1 #HV01 #T1 #HVT1 +#KX #VX #H #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02 +lapply (ldrop_mono … H … HLK0) -H #H destruct +elim (lpr_ldrop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1 +elim (lpr_inv_pair1 … H1) -H1 #K1 #W1 #HK01 #_ #H destruct +lapply (ldrop_fwd_ldrop2 … HLK1) -W1 #HLK1 +elim (lpss_ldrop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2 +elim (lpss_inv_pair1 … H2) -H2 #K2 #W2 #HK02 #_ #H destruct +lapply (ldrop_fwd_ldrop2 … HLK2) -W2 #HLK2 +lapply (ldrop_pair2_fwd_fw … HLK0 (#i)) -HLK0 #HLK0 +elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2 +elim (lift_total V 0 (i+1)) #T #HVT +lapply (cpss_lift … HV1 … HLK1 … HVT1 … HVT) -K1 -V1 +lapply (cpr_lift … HV2 … HLK2 … HVT2 … HVT) -K2 -V2 -V /2 width=3/ +qed-. + +fact cpr_cpss_conf_lpr_lpss_bind_bind: + ∀a,I,L0,V0,T0. ( + ∀L,T.♯{L,T} < ♯{L0,ⓑ{a,I}V0.T0} → + ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 → + ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 → + ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0 + ) → + ∀V1. L0 ⊢ V0 ➡ V1 → ∀T1. L0.ⓑ{I}V0 ⊢ T0 ➡ T1 → + ∀V2. L0 ⊢ V0 ▶* V2 → ∀T2. L0.ⓑ{I}V0 ⊢ T0 ▶* T2 → + ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 → + ∃∃T. L1 ⊢ ⓑ{a,I}V1.T1 ▶* T & L2 ⊢ ⓑ{a,I}V2.T2 ➡ T. +#a #I #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01 +#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02 +elim (IH … HV01 … HV02 … HL01 … HL02) // +elim (IH … HT01 … HT02 (L1.ⓑ{I}V1) … (L2.ⓑ{I}V2)) -IH // /2 width=1/ /3 width=5/ +qed-. + +fact cpr_cpss_conf_lpr_lpss_bind_zeta: + ∀L0,V0,T0. ( + ∀L,T.♯{L,T} < ♯{L0,+ⓓV0.T0} → + ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 → + ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 → + ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0 + ) → + ∀T1. L0.ⓓV0 ⊢ T0 ➡ T1 → ∀X1. ⇧[O, 1] X1 ≡ T1 → + ∀V2. L0 ⊢ V0 ▶* V2 → ∀T2. L0.ⓓV0 ⊢ T0 ▶* T2 → + ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 → + ∃∃T. L1 ⊢ X1 ▶* T & L2 ⊢ +ⓓV2.T2 ➡ T. +#L0 #V0 #T0 #IH #T1 #HT01 #X1 #HXT1 +#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02 +elim (IH … HT01 … HT02 (L1.ⓓV2) … (L2.ⓓV2)) -IH -HT01 -HT02 // /2 width=1/ /3 width=1/ -L0 -V0 -T0 #T #HT1 #HT2 +elim (cpss_inv_lift1 … HT1 L1 … HXT1) -T1 /2 width=1/ /3 width=9/ +qed-. + +fact cpr_cpss_conf_lpr_lpss_flat_flat: + ∀I,L0,V0,T0. ( + ∀L,T.♯{L,T} < ♯{L0,ⓕ{I}V0.T0} → + ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 → + ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 → + ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0 + ) → + ∀V1. L0 ⊢ V0 ➡ V1 → ∀T1. L0 ⊢ T0 ➡ T1 → + ∀V2. L0 ⊢ V0 ▶* V2 → ∀T2. L0 ⊢ T0 ▶* T2 → + ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 → + ∃∃T. L1 ⊢ ⓕ{I}V1.T1 ▶* T & L2 ⊢ ⓕ{I}V2.T2 ➡ T. +#I #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01 +#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02 +elim (IH … HV01 … HV02 … HL01 … HL02) // +elim (IH … HT01 … HT02 … HL01 … HL02) // /3 width=5/ +qed-. + +fact cpr_cpss_conf_lpr_lpss_flat_tau: + ∀L0,V0,T0. ( + ∀L,T.♯{L,T} < ♯{L0,ⓝV0.T0} → + ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 → + ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 → + ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0 + ) → + ∀T1. L0 ⊢ T0 ➡ T1 → ∀V2,T2. L0 ⊢ T0 ▶* T2 → + ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 → + ∃∃T. L1 ⊢ T1 ▶* T & L2 ⊢ ⓝV2.T2 ➡ T. +#L0 #V0 #T0 #IH #T1 #HT01 +#V2 #T2 #HT02 #L1 #HL01 #L2 #HL02 +elim (IH … HT01 … HT02 … HL01 … HL02) // -L0 -V0 -T0 /3 width=3/ +qed-. + +fact cpr_cpss_conf_lpr_lpss_flat_beta: + ∀a,L0,V0,W0,T0. ( + ∀L,T.♯{L,T} < ♯{L0,ⓐV0.ⓛ{a}W0.T0} → + ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 → + ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 → + ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0 + ) → + ∀V1. L0 ⊢ V0 ➡ V1 → ∀T1. L0.ⓛW0 ⊢ T0 ➡ T1 → + ∀V2. L0 ⊢ V0 ▶* V2 → ∀T2. L0 ⊢ ⓛ{a}W0.T0 ▶* T2 → + ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 → + ∃∃T. L1 ⊢ ⓓ{a}V1.T1 ▶* T & L2 ⊢ ⓐV2.T2 ➡ T. +#a #L0 #V0 #W0 #T0 #IH #V1 #HV01 #T1 #HT01 +#V2 #HV02 #X #H #L1 #HL01 #L2 #HL02 +elim (cpss_inv_bind1 … H) -H #W2 #T2 #HW02 #HT02 #H destruct +elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1/ #V #HV1 #HV2 +elim (IH … HT01 … HT02 (L1.ⓛW2) … (L2.ⓛW2)) /2 width=1/ /3 width=1/ -L0 -V0 -W0 -T0 #T #HT1 #HT2 +lapply (cpss_lsubr_trans … HT1 (L1.ⓓV1) ?) -HT1 /2 width=1/ /3 width=5/ +qed-. + +fact cpr_cpss_conf_lpr_lpss_flat_theta: + ∀a,L0,V0,W0,T0. ( + ∀L,T.♯{L,T} < ♯{L0,ⓐV0.ⓓ{a}W0.T0} → + ∀T1. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ▶* T2 → + ∀L1. L ⊢ ➡ L1 → ∀L2. L ⊢ ▶* L2 → + ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ➡ T0 + ) → + ∀V1. L0 ⊢ V0 ➡ V1 → ∀U1. ⇧[O, 1] V1 ≡ U1 → + ∀W1. L0 ⊢ W0 ➡ W1 → ∀T1. L0.ⓓW0 ⊢ T0 ➡ T1 → + ∀V2. L0 ⊢ V0 ▶* V2 → ∀T2. L0 ⊢ ⓓ{a}W0.T0 ▶* T2 → + ∀L1. L0 ⊢ ➡ L1 → ∀L2. L0 ⊢ ▶* L2 → + ∃∃T. L1 ⊢ ⓓ{a}W1.ⓐU1.T1 ▶* T & L2 ⊢ ⓐV2.T2 ➡ T. +#a #L0 #V0 #W0 #T0 #IH #V1 #HV01 #U1 #HVU1 #W1 #HW01 #T1 #HT01 +#V2 #HV02 #X #H #L1 #HL01 #L2 #HL02 +elim (IH … HV01 … HV02 … HL01 … HL02) -HV01 -HV02 /2 width=1/ #V #HV1 #HV2 +elim (lift_total V 0 1) #U #HVU +lapply (cpss_lift … HV1 (L1.ⓓW1) … HVU1 … HVU) -HVU1 /2 width=1/ #HU1 +elim (cpss_inv_bind1 … H) -H #W2 #T2 #HW02 #HT02 #H destruct +elim (IH … HW01 … HW02 … HL01 … HL02) /2 width=1/ +elim (IH … HT01 … HT02 (L1.ⓓW1) … (L2.ⓓW2)) /2 width=1/ -L0 -V0 -W0 -T0 +/4 width=9 by ex2_intro, cpr_theta, cpss_bind, cpss_flat/ (**) (* auto too slow without trace *) +qed-. + +lemma cpr_cpss_conf_lpr_lpss: lpx_sn_confluent cpr cpss. +#L0 #T0 @(f2_ind … fw … L0 T0) -L0 -T0 #n #IH #L0 * [|*] +[ #I0 #Hn #T1 #H1 #T2 #H2 #L1 #HL01 #L2 #HL02 destruct + elim (cpr_inv_atom1 … H1) -H1 + elim (cpss_inv_atom1 … H2) -H2 + [ #H2 #H1 destruct + /2 width=1 by cpr_cpss_conf_lpr_lpss_atom_atom/ + | * #K0 #V0 #V2 #i2 #HLK0 #HV02 #HVT2 #H2 #H1 destruct + /3 width=10 by cpr_cpss_conf_lpr_lpss_atom_delta/ + | #H2 * #K0 #V0 #V1 #i1 #HLK0 #HV01 #HVT1 #H1 destruct + /3 width=10 by cpr_cpss_conf_lpr_lpss_delta_atom/ + | * #X #Y #V2 #z #H #HV02 #HVT2 #H2 + * #K0 #V0 #V1 #i #HLK0 #HV01 #HVT1 #H1 destruct + /3 width=17 by cpr_cpss_conf_lpr_lpss_delta_delta/ + ] +| #a #I #V0 #T0 #Hn #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct + elim (cpss_inv_bind1 … H2) -H2 #V2 #T2 #HV02 #HT02 #H2 + elim (cpr_inv_bind1 … H1) -H1 * + [ #V1 #T1 #HV01 #HT01 #H1 + | #T1 #HT01 #HXT1 #H11 #H12 + ] destruct + [ /3 width=10 by cpr_cpss_conf_lpr_lpss_bind_bind/ + | /3 width=11 by cpr_cpss_conf_lpr_lpss_bind_zeta/ + ] +| #I #V0 #T0 #Hn #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct + elim (cpss_inv_flat1 … H2) -H2 #V2 #T2 #HV02 #HT02 #H2 + elim (cpr_inv_flat1 … H1) -H1 * + [ #V1 #T1 #HV01 #HT01 #H1 + | #HX1 #H1 + | #a1 #V1 #Y1 #Z1 #T1 #HV01 #HZT1 #H11 #H12 #H13 + | #a1 #V1 #U1 #Y1 #W1 #Z1 #T1 #HV01 #HVU1 #HYW1 #HZT1 #H11 #H12 #H13 + ] destruct + [ /3 width=10 by cpr_cpss_conf_lpr_lpss_flat_flat/ + | /3 width=8 by cpr_cpss_conf_lpr_lpss_flat_tau/ + | /3 width=11 by cpr_cpss_conf_lpr_lpss_flat_beta/ + | /3 width=14 by cpr_cpss_conf_lpr_lpss_flat_theta/ + ] +] +qed-. + +(* Basic_1: includes: pr0_subst1 *) +lemma cpr_cpss_conf: ∀L. confluent2 … (cpr L) (cpss L). +/2 width=6 by cpr_cpss_conf_lpr_lpss/ qed-. + +lemma cpr_lpss_conf_dx: ∀L0,T0,T1. L0 ⊢ T0 ➡ T1 → ∀L1. L0 ⊢ ▶* L1 → + ∃∃T. L1 ⊢ T1 ▶* T & L1 ⊢ T0 ➡ T. +#L0 #T0 #T1 #HT01 #L1 #HL01 +elim (cpr_cpss_conf_lpr_lpss … HT01 T0 … L1 … HL01) // /2 width=1/ -L0 /2 width=3/ +qed-. + +lemma cpr_lpss_conf_sn: ∀L0,T0,T1. L0 ⊢ T0 ➡ T1 → ∀L1. L0 ⊢ ▶* L1 → + ∃∃T. L0 ⊢ T1 ▶* T & L1 ⊢ T0 ➡ T. +#L0 #T0 #T1 #HT01 #L1 #HL01 +elim (cpr_cpss_conf_lpr_lpss … HT01 T0 … L0 … HL01) // -HT01 -HL01 /2 width=3/ +qed-. + +(* Basic_1: includes: pr0_subst1_fwd *) +lemma lpr_cpss_conf: ∀L0,T0,T1. L0 ⊢ T0 ▶* T1 → ∀L1. L0 ⊢ ➡ L1 → + ∃∃T. L1 ⊢ T0 ▶* T & L0 ⊢ T1 ➡ T. +#L0 #T0 #T1 #HT01 #L1 #HL01 +elim (cpr_cpss_conf_lpr_lpss ?? T0 … HT01 … HL01 L0) // -HT01 -HL01 /2 width=3/ +qed-. (* Properties on sn parallel substitution on local environments *************) diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/lpss_cpss.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/lpss_cpss.ma index 2a418d759..c9c17fa19 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/substitution/lpss_cpss.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/lpss_cpss.ma @@ -13,133 +13,12 @@ (**************************************************************************) include "basic_2/grammar/lpx_sn_lpx_sn.ma". -include "basic_2/relocation/fsup.ma". include "basic_2/substitution/lpss_ldrop.ma". (* SN PARALLEL SUBSTITUTION FOR LOCAL ENVIRONMENTS **************************) (* Main properties on context-sensitive parallel substitution for terms *****) -fact cpss_conf_lpss_atom_atom: - ∀I,L1,L2. ∃∃T. L1 ⊢ ⓪{I} ▶* T & L2 ⊢ ⓪{I} ▶* T. -/2 width=3/ qed-. - -fact cpss_conf_lpss_atom_delta: - ∀L0,i. ( - ∀L,T.♯{L, T} < ♯{L0, #i} → - ∀T1. L ⊢ T ▶* T1 → ∀T2. L ⊢ T ▶* T2 → - ∀L1. L ⊢ ▶* L1 → ∀L2. L ⊢ ▶* L2 → - ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ▶* T0 - ) → - ∀K0,V0. ⇩[O, i] L0 ≡ K0.ⓓV0 → - ∀V2. K0 ⊢ V0 ▶* V2 → ∀T2. ⇧[O, i + 1] V2 ≡ T2 → - ∀L1. L0 ⊢ ▶* L1 → ∀L2. L0 ⊢ ▶* L2 → - ∃∃T. L1 ⊢ #i ▶* T & L2 ⊢ T2 ▶* T. -#L0 #i #IH #K0 #V0 #HLK0 #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02 -elim (lpss_ldrop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1 -elim (lpss_inv_pair1 … H1) -H1 #K1 #V1 #HK01 #HV01 #H destruct -elim (lpss_ldrop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2 -elim (lpss_inv_pair1 … H2) -H2 #K2 #W2 #HK02 #_ #H destruct -lapply (ldrop_fwd_ldrop2 … HLK2) -W2 #HLK2 -lapply (ldrop_pair2_fwd_fw … HLK0 (#i)) -HLK0 #HLK0 -elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2 -elim (lift_total V 0 (i+1)) #T #HVT -lapply (cpss_lift … HV2 … HLK2 … HVT2 … HVT) -K2 -V2 /3 width=6/ -qed-. - -fact cpss_conf_lpss_delta_delta: - ∀L0,i. ( - ∀L,T.♯{L, T} < ♯{L0, #i} → - ∀T1. L ⊢ T ▶* T1 → ∀T2. L ⊢ T ▶* T2 → - ∀L1. L ⊢ ▶* L1 → ∀L2. L ⊢ ▶* L2 → - ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ▶* T0 - ) → - ∀K0,V0. ⇩[O, i] L0 ≡ K0.ⓓV0 → - ∀V1. K0 ⊢ V0 ▶* V1 → ∀T1. ⇧[O, i + 1] V1 ≡ T1 → - ∀KX,VX. ⇩[O, i] L0 ≡ KX.ⓓVX → - ∀V2. KX ⊢ VX ▶* V2 → ∀T2. ⇧[O, i + 1] V2 ≡ T2 → - ∀L1. L0 ⊢ ▶* L1 → ∀L2. L0 ⊢ ▶* L2 → - ∃∃T. L1 ⊢ T1 ▶* T & L2 ⊢ T2 ▶* T. -#L0 #i #IH #K0 #V0 #HLK0 #V1 #HV01 #T1 #HVT1 -#KX #VX #H #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02 -lapply (ldrop_mono … H … HLK0) -H #H destruct -elim (lpss_ldrop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1 -elim (lpss_inv_pair1 … H1) -H1 #K1 #W1 #HK01 #_ #H destruct -lapply (ldrop_fwd_ldrop2 … HLK1) -W1 #HLK1 -elim (lpss_ldrop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2 -elim (lpss_inv_pair1 … H2) -H2 #K2 #W2 #HK02 #_ #H destruct -lapply (ldrop_fwd_ldrop2 … HLK2) -W2 #HLK2 -lapply (ldrop_pair2_fwd_fw … HLK0 (#i)) -HLK0 #HLK0 -elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2 -elim (lift_total V 0 (i+1)) #T #HVT -lapply (cpss_lift … HV1 … HLK1 … HVT1 … HVT) -K1 -V1 -lapply (cpss_lift … HV2 … HLK2 … HVT2 … HVT) -K2 -V2 -V /2 width=3/ -qed-. - -fact cpss_conf_lpss_bind_bind: - ∀a,I,L0,V0,T0. ( - ∀L,T.♯{L,T} < ♯{L0,ⓑ{a,I}V0.T0} → - ∀T1. L ⊢ T ▶* T1 → ∀T2. L ⊢ T ▶* T2 → - ∀L1. L ⊢ ▶* L1 → ∀L2. L ⊢ ▶* L2 → - ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ▶* T0 - ) → - ∀V1. L0 ⊢ V0 ▶* V1 → ∀T1. L0.ⓑ{I}V0 ⊢ T0 ▶* T1 → - ∀V2. L0 ⊢ V0 ▶* V2 → ∀T2. L0.ⓑ{I}V0 ⊢ T0 ▶* T2 → - ∀L1. L0 ⊢ ▶* L1 → ∀L2. L0 ⊢ ▶* L2 → - ∃∃T. L1 ⊢ ⓑ{a,I}V1.T1 ▶* T & L2 ⊢ ⓑ{a,I}V2.T2 ▶* T. -#a #I #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01 -#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02 -elim (IH … HV01 … HV02 … HL01 … HL02) // -elim (IH … HT01 … HT02 (L1.ⓑ{I}V1) … (L2.ⓑ{I}V2)) -IH // /2 width=1/ /3 width=5/ -qed-. - -fact cpss_conf_lpss_flat_flat: - ∀I,L0,V0,T0. ( - ∀L,T.♯{L,T} < ♯{L0,ⓕ{I}V0.T0} → - ∀T1. L ⊢ T ▶* T1 → ∀T2. L ⊢ T ▶* T2 → - ∀L1. L ⊢ ▶* L1 → ∀L2. L ⊢ ▶* L2 → - ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ▶* T0 - ) → - ∀V1. L0 ⊢ V0 ▶* V1 → ∀T1. L0 ⊢ T0 ▶* T1 → - ∀V2. L0 ⊢ V0 ▶* V2 → ∀T2. L0 ⊢ T0 ▶* T2 → - ∀L1. L0 ⊢ ▶* L1 → ∀L2. L0 ⊢ ▶* L2 → - ∃∃T. L1 ⊢ ⓕ{I}V1.T1 ▶* T & L2 ⊢ ⓕ{I}V2.T2 ▶* T. -#I #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01 -#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02 -elim (IH … HV01 … HV02 … HL01 … HL02) // -elim (IH … HT01 … HT02 … HL01 … HL02) // /3 width=5/ -qed-. - -theorem cpss_conf_lpss: lpx_sn_confluent cpss cpss. -#L0 #T0 @(f2_ind … fw … L0 T0) -L0 -T0 #n #IH #L0 * [|*] -[ #I0 #Hn #T1 #H1 #T2 #H2 #L1 #HL01 #L2 #HL02 destruct - elim (cpss_inv_atom1 … H1) -H1 - elim (cpss_inv_atom1 … H2) -H2 - [ #H2 #H1 destruct - /2 width=1 by cpss_conf_lpss_atom_atom/ - | * #K0 #V0 #V2 #i2 #HLK0 #HV02 #HVT2 #H2 #H1 destruct - /3 width=10 by cpss_conf_lpss_atom_delta/ - | #H2 * #K0 #V0 #V1 #i1 #HLK0 #HV01 #HVT1 #H1 destruct - /4 width=10 by ex2_commute, cpss_conf_lpss_atom_delta/ - | * #X #Y #V2 #z #H #HV02 #HVT2 #H2 - * #K0 #V0 #V1 #i #HLK0 #HV01 #HVT1 #H1 destruct - /3 width=17 by cpss_conf_lpss_delta_delta/ - ] -| #a #I #V0 #T0 #Hn #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct - elim (cpss_inv_bind1 … H1) -H1 #V1 #T1 #HV01 #HT01 #H destruct - elim (cpss_inv_bind1 … H2) -H2 #V2 #T2 #HV02 #HT02 #H destruct - /3 width=10 by cpss_conf_lpss_bind_bind/ -| #I #V0 #T0 #Hn #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct - elim (cpss_inv_flat1 … H1) -H1 #V1 #T1 #HV01 #HT01 #H destruct - elim (cpss_inv_flat1 … H2) -H2 #V2 #T2 #HV02 #HT02 #H destruct - /3 width=10 by cpss_conf_lpss_flat_flat/ -] -qed-. - -(* Basic_1: was only: subst1_confluence_eq *) -theorem cpss_conf: ∀L. confluent … (cpss L). -/2 width=6 by cpss_conf_lpss/ qed-. - theorem cpss_trans_lpss: lpx_sn_transitive cpss cpss. #L1 #T1 @(f2_ind … fw … L1 T1) -L1 -T1 #n #IH #L1 * [|*] [ #I #Hn #T #H1 #L2 #HL12 #T2 #HT2 destruct @@ -174,30 +53,7 @@ theorem cpss_trans: ∀L. Transitive … (cpss L). (* Properties on context-sensitive parallel substitution for terms **********) -(* Basic_1: was only: subst1_subst1_back *) -lemma lpss_cpss_conf_dx: ∀L0,T0,T1. L0 ⊢ T0 ▶* T1 → ∀L1. L0 ⊢ ▶* L1 → - ∃∃T. L1 ⊢ T0 ▶* T & L1 ⊢ T1 ▶* T. -#L0 #T0 #T1 #HT01 #L1 #HL01 -elim (cpss_conf_lpss … HT01 T0 … HL01 … HL01) // -L0 /2 width=3/ -qed-. - -lemma lpss_cpss_conf_sn: ∀L0,T0,T1. L0 ⊢ T0 ▶* T1 → ∀L1. L0 ⊢ ▶* L1 → - ∃∃T. L1 ⊢ T0 ▶* T & L0 ⊢ T1 ▶* T. -#L0 #T0 #T1 #HT01 #L1 #HL01 -elim (cpss_conf_lpss … HT01 T0 … L0 … HL01) // -HT01 -HL01 /2 width=3/ -qed-. - (* Basic_1: was only: subst1_subst1 *) lemma lpss_cpss_trans: ∀L1,L2. L1 ⊢ ▶* L2 → ∀T1,T2. L2 ⊢ T1 ▶* T2 → L1 ⊢ T1 ▶* T2. /2 width=5 by cpss_trans_lpss/ qed-. - -lemma fsup_cpss_trans: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ → ∀U2. L2 ⊢ T2 ▶* U2 → - ∃∃L,U1. L1 ⊢ ▶* L & L ⊢ T1 ▶* U1 & ⦃L, U1⦄ ⊃ ⦃L2, U2⦄. -#L1 #L2 #T1 #T2 #H elim H -L1 -L2 -T1 -T2 [1,2,3,4,5: /3 width=5/ ] -#L1 #K1 #K2 #T1 #T2 #U1 #d #e #HLK1 #HTU1 #_ #IHT12 #U2 #HTU2 -elim (IHT12 … HTU2) -IHT12 -HTU2 #K #T #HK1 #HT1 #HT2 -elim (lift_total T d e) #U #HTU -elim (ldrop_lpss_trans … HLK1 … HK1) -HLK1 -HK1 #L2 #HL12 #HL2K -lapply (cpss_lift … HT1 … HL2K … HTU1 … HTU) -HT1 -HTU1 /3 width=11/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/lpss_ldrop.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/lpss_ldrop.ma index 044293b27..17fa5b45f 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/substitution/lpss_ldrop.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/lpss_ldrop.ma @@ -12,6 +12,7 @@ (* *) (**************************************************************************) +include "basic_2/relocation/fsup.ma". include "basic_2/relocation/ldrop_lpx_sn.ma". include "basic_2/substitution/cpss_lift.ma". include "basic_2/substitution/lpss.ma". @@ -28,3 +29,15 @@ lemma ldrop_lpss_trans: dedropable_sn lpss. lemma lpss_ldrop_trans_O1: dropable_dx lpss. /2 width=3 by lpx_sn_dropable/ qed-. + +(* Properties on context-sensitive parallel substitution for terms **********) + +lemma fsup_cpss_trans: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ → ∀U2. L2 ⊢ T2 ▶* U2 → + ∃∃L,U1. L1 ⊢ ▶* L & L ⊢ T1 ▶* U1 & ⦃L, U1⦄ ⊃ ⦃L2, U2⦄. +#L1 #L2 #T1 #T2 #H elim H -L1 -L2 -T1 -T2 [1,2,3,4,5: /3 width=5/ ] +#L1 #K1 #K2 #T1 #T2 #U1 #d #e #HLK1 #HTU1 #_ #IHT12 #U2 #HTU2 +elim (IHT12 … HTU2) -IHT12 -HTU2 #K #T #HK1 #HT1 #HT2 +elim (lift_total T d e) #U #HTU +elim (ldrop_lpss_trans … HLK1 … HK1) -HLK1 -HK1 #L2 #HL12 #HL2K +lapply (cpss_lift … HT1 … HL2K … HTU1 … HTU) -HT1 -HTU1 /3 width=11/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/lpss_lpss.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/lpss_lpss.ma index 06223fbee..ad972cd4a 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/substitution/lpss_lpss.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/lpss_lpss.ma @@ -16,6 +16,143 @@ include "basic_2/substitution/lpss_cpss.ma". (* SN PARALLEL SUBSTITUTION ON LOCAL ENVIRONMENTS ***************************) +(* Main properties on context-sensitive parallel substitution for terms *****) + +fact cpss_conf_lpss_atom_atom: + ∀I,L1,L2. ∃∃T. L1 ⊢ ⓪{I} ▶* T & L2 ⊢ ⓪{I} ▶* T. +/2 width=3/ qed-. + +fact cpss_conf_lpss_atom_delta: + ∀L0,i. ( + ∀L,T.♯{L, T} < ♯{L0, #i} → + ∀T1. L ⊢ T ▶* T1 → ∀T2. L ⊢ T ▶* T2 → + ∀L1. L ⊢ ▶* L1 → ∀L2. L ⊢ ▶* L2 → + ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ▶* T0 + ) → + ∀K0,V0. ⇩[O, i] L0 ≡ K0.ⓓV0 → + ∀V2. K0 ⊢ V0 ▶* V2 → ∀T2. ⇧[O, i + 1] V2 ≡ T2 → + ∀L1. L0 ⊢ ▶* L1 → ∀L2. L0 ⊢ ▶* L2 → + ∃∃T. L1 ⊢ #i ▶* T & L2 ⊢ T2 ▶* T. +#L0 #i #IH #K0 #V0 #HLK0 #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02 +elim (lpss_ldrop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1 +elim (lpss_inv_pair1 … H1) -H1 #K1 #V1 #HK01 #HV01 #H destruct +elim (lpss_ldrop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2 +elim (lpss_inv_pair1 … H2) -H2 #K2 #W2 #HK02 #_ #H destruct +lapply (ldrop_fwd_ldrop2 … HLK2) -W2 #HLK2 +lapply (ldrop_pair2_fwd_fw … HLK0 (#i)) -HLK0 #HLK0 +elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2 +elim (lift_total V 0 (i+1)) #T #HVT +lapply (cpss_lift … HV2 … HLK2 … HVT2 … HVT) -K2 -V2 /3 width=6/ +qed-. + +fact cpss_conf_lpss_delta_delta: + ∀L0,i. ( + ∀L,T.♯{L, T} < ♯{L0, #i} → + ∀T1. L ⊢ T ▶* T1 → ∀T2. L ⊢ T ▶* T2 → + ∀L1. L ⊢ ▶* L1 → ∀L2. L ⊢ ▶* L2 → + ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ▶* T0 + ) → + ∀K0,V0. ⇩[O, i] L0 ≡ K0.ⓓV0 → + ∀V1. K0 ⊢ V0 ▶* V1 → ∀T1. ⇧[O, i + 1] V1 ≡ T1 → + ∀KX,VX. ⇩[O, i] L0 ≡ KX.ⓓVX → + ∀V2. KX ⊢ VX ▶* V2 → ∀T2. ⇧[O, i + 1] V2 ≡ T2 → + ∀L1. L0 ⊢ ▶* L1 → ∀L2. L0 ⊢ ▶* L2 → + ∃∃T. L1 ⊢ T1 ▶* T & L2 ⊢ T2 ▶* T. +#L0 #i #IH #K0 #V0 #HLK0 #V1 #HV01 #T1 #HVT1 +#KX #VX #H #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02 +lapply (ldrop_mono … H … HLK0) -H #H destruct +elim (lpss_ldrop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1 +elim (lpss_inv_pair1 … H1) -H1 #K1 #W1 #HK01 #_ #H destruct +lapply (ldrop_fwd_ldrop2 … HLK1) -W1 #HLK1 +elim (lpss_ldrop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2 +elim (lpss_inv_pair1 … H2) -H2 #K2 #W2 #HK02 #_ #H destruct +lapply (ldrop_fwd_ldrop2 … HLK2) -W2 #HLK2 +lapply (ldrop_pair2_fwd_fw … HLK0 (#i)) -HLK0 #HLK0 +elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2 +elim (lift_total V 0 (i+1)) #T #HVT +lapply (cpss_lift … HV1 … HLK1 … HVT1 … HVT) -K1 -V1 +lapply (cpss_lift … HV2 … HLK2 … HVT2 … HVT) -K2 -V2 -V /2 width=3/ +qed-. + +fact cpss_conf_lpss_bind_bind: + ∀a,I,L0,V0,T0. ( + ∀L,T.♯{L,T} < ♯{L0,ⓑ{a,I}V0.T0} → + ∀T1. L ⊢ T ▶* T1 → ∀T2. L ⊢ T ▶* T2 → + ∀L1. L ⊢ ▶* L1 → ∀L2. L ⊢ ▶* L2 → + ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ▶* T0 + ) → + ∀V1. L0 ⊢ V0 ▶* V1 → ∀T1. L0.ⓑ{I}V0 ⊢ T0 ▶* T1 → + ∀V2. L0 ⊢ V0 ▶* V2 → ∀T2. L0.ⓑ{I}V0 ⊢ T0 ▶* T2 → + ∀L1. L0 ⊢ ▶* L1 → ∀L2. L0 ⊢ ▶* L2 → + ∃∃T. L1 ⊢ ⓑ{a,I}V1.T1 ▶* T & L2 ⊢ ⓑ{a,I}V2.T2 ▶* T. +#a #I #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01 +#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02 +elim (IH … HV01 … HV02 … HL01 … HL02) // +elim (IH … HT01 … HT02 (L1.ⓑ{I}V1) … (L2.ⓑ{I}V2)) -IH // /2 width=1/ /3 width=5/ +qed-. + +fact cpss_conf_lpss_flat_flat: + ∀I,L0,V0,T0. ( + ∀L,T.♯{L,T} < ♯{L0,ⓕ{I}V0.T0} → + ∀T1. L ⊢ T ▶* T1 → ∀T2. L ⊢ T ▶* T2 → + ∀L1. L ⊢ ▶* L1 → ∀L2. L ⊢ ▶* L2 → + ∃∃T0. L1 ⊢ T1 ▶* T0 & L2 ⊢ T2 ▶* T0 + ) → + ∀V1. L0 ⊢ V0 ▶* V1 → ∀T1. L0 ⊢ T0 ▶* T1 → + ∀V2. L0 ⊢ V0 ▶* V2 → ∀T2. L0 ⊢ T0 ▶* T2 → + ∀L1. L0 ⊢ ▶* L1 → ∀L2. L0 ⊢ ▶* L2 → + ∃∃T. L1 ⊢ ⓕ{I}V1.T1 ▶* T & L2 ⊢ ⓕ{I}V2.T2 ▶* T. +#I #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01 +#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02 +elim (IH … HV01 … HV02 … HL01 … HL02) // +elim (IH … HT01 … HT02 … HL01 … HL02) // /3 width=5/ +qed-. + +theorem cpss_conf_lpss: lpx_sn_confluent cpss cpss. +#L0 #T0 @(f2_ind … fw … L0 T0) -L0 -T0 #n #IH #L0 * [|*] +[ #I0 #Hn #T1 #H1 #T2 #H2 #L1 #HL01 #L2 #HL02 destruct + elim (cpss_inv_atom1 … H1) -H1 + elim (cpss_inv_atom1 … H2) -H2 + [ #H2 #H1 destruct + /2 width=1 by cpss_conf_lpss_atom_atom/ + | * #K0 #V0 #V2 #i2 #HLK0 #HV02 #HVT2 #H2 #H1 destruct + /3 width=10 by cpss_conf_lpss_atom_delta/ + | #H2 * #K0 #V0 #V1 #i1 #HLK0 #HV01 #HVT1 #H1 destruct + /4 width=10 by ex2_commute, cpss_conf_lpss_atom_delta/ + | * #X #Y #V2 #z #H #HV02 #HVT2 #H2 + * #K0 #V0 #V1 #i #HLK0 #HV01 #HVT1 #H1 destruct + /3 width=17 by cpss_conf_lpss_delta_delta/ + ] +| #a #I #V0 #T0 #Hn #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct + elim (cpss_inv_bind1 … H1) -H1 #V1 #T1 #HV01 #HT01 #H destruct + elim (cpss_inv_bind1 … H2) -H2 #V2 #T2 #HV02 #HT02 #H destruct + /3 width=10 by cpss_conf_lpss_bind_bind/ +| #I #V0 #T0 #Hn #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct + elim (cpss_inv_flat1 … H1) -H1 #V1 #T1 #HV01 #HT01 #H destruct + elim (cpss_inv_flat1 … H2) -H2 #V2 #T2 #HV02 #HT02 #H destruct + /3 width=10 by cpss_conf_lpss_flat_flat/ +] +qed-. + +(* Basic_1: was only: subst1_confluence_eq *) +theorem cpss_conf: ∀L. confluent … (cpss L). +/2 width=6 by cpss_conf_lpss/ qed-. + +(* Properties on context-sensitive parallel substitution for terms **********) + +(* Basic_1: was only: subst1_subst1_back *) +lemma lpss_cpss_conf_dx: ∀L0,T0,T1. L0 ⊢ T0 ▶* T1 → ∀L1. L0 ⊢ ▶* L1 → + ∃∃T. L1 ⊢ T0 ▶* T & L1 ⊢ T1 ▶* T. +#L0 #T0 #T1 #HT01 #L1 #HL01 +elim (cpss_conf_lpss … HT01 T0 … HL01 … HL01) // -L0 /2 width=3/ +qed-. + +lemma lpss_cpss_conf_sn: ∀L0,T0,T1. L0 ⊢ T0 ▶* T1 → ∀L1. L0 ⊢ ▶* L1 → + ∃∃T. L1 ⊢ T0 ▶* T & L0 ⊢ T1 ▶* T. +#L0 #T0 #T1 #HT01 #L1 #HL01 +elim (cpss_conf_lpss … HT01 T0 … L0 … HL01) // -HT01 -HL01 /2 width=3/ +qed-. + (* Main properties **********************************************************) theorem lpss_conf: confluent … lpss. diff --git a/matita/matita/contribs/lambdadelta/basic_2/unfold/lpqs_cpqs.ma b/matita/matita/contribs/lambdadelta/basic_2/unfold/lpqs_cpqs.ma index d546005bb..fb94a49de 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/unfold/lpqs_cpqs.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/unfold/lpqs_cpqs.ma @@ -13,218 +13,12 @@ (**************************************************************************) include "basic_2/grammar/lpx_sn_lpx_sn.ma". -include "basic_2/relocation/fsup.ma". include "basic_2/unfold/lpqs_ldrop.ma". (* SN RESTRICTED PARALLEL COMPUTATION FOR LOCAL ENVIRONMENTS ****************) (* Main properties on context-sensitive rest parallel computation for terms *) -fact cpqs_conf_lpqs_atom_atom: - ∀I,L1,L2. ∃∃T. L1 ⊢ ⓪{I} ➤* T & L2 ⊢ ⓪{I} ➤* T. -/2 width=3/ qed-. - -fact cpqs_conf_lpqs_atom_delta: - ∀L0,i. ( - ∀L,T.♯{L, T} < ♯{L0, #i} → - ∀T1. L ⊢ T ➤* T1 → ∀T2. L ⊢ T ➤* T2 → - ∀L1. L ⊢ ➤* L1 → ∀L2. L ⊢ ➤* L2 → - ∃∃T0. L1 ⊢ T1 ➤* T0 & L2 ⊢ T2 ➤* T0 - ) → - ∀K0,V0. ⇩[O, i] L0 ≡ K0.ⓓV0 → - ∀V2. K0 ⊢ V0 ➤* V2 → ∀T2. ⇧[O, i + 1] V2 ≡ T2 → - ∀L1. L0 ⊢ ➤* L1 → ∀L2. L0 ⊢ ➤* L2 → - ∃∃T. L1 ⊢ #i ➤* T & L2 ⊢ T2 ➤* T. -#L0 #i #IH #K0 #V0 #HLK0 #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02 -elim (lpqs_ldrop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1 -elim (lpqs_inv_pair1 … H1) -H1 #K1 #V1 #HK01 #HV01 #H destruct -elim (lpqs_ldrop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2 -elim (lpqs_inv_pair1 … H2) -H2 #K2 #W2 #HK02 #_ #H destruct -lapply (ldrop_fwd_ldrop2 … HLK2) -W2 #HLK2 -lapply (ldrop_pair2_fwd_fw … HLK0 (#i)) -HLK0 #HLK0 -elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2 -elim (lift_total V 0 (i+1)) #T #HVT -lapply (cpqs_lift … HV2 … HLK2 … HVT2 … HVT) -K2 -V2 /3 width=6/ -qed-. - -fact cpqs_conf_lpqs_delta_delta: - ∀L0,i. ( - ∀L,T.♯{L, T} < ♯{L0, #i} → - ∀T1. L ⊢ T ➤* T1 → ∀T2. L ⊢ T ➤* T2 → - ∀L1. L ⊢ ➤* L1 → ∀L2. L ⊢ ➤* L2 → - ∃∃T0. L1 ⊢ T1 ➤* T0 & L2 ⊢ T2 ➤* T0 - ) → - ∀K0,V0. ⇩[O, i] L0 ≡ K0.ⓓV0 → - ∀V1. K0 ⊢ V0 ➤* V1 → ∀T1. ⇧[O, i + 1] V1 ≡ T1 → - ∀KX,VX. ⇩[O, i] L0 ≡ KX.ⓓVX → - ∀V2. KX ⊢ VX ➤* V2 → ∀T2. ⇧[O, i + 1] V2 ≡ T2 → - ∀L1. L0 ⊢ ➤* L1 → ∀L2. L0 ⊢ ➤* L2 → - ∃∃T. L1 ⊢ T1 ➤* T & L2 ⊢ T2 ➤* T. -#L0 #i #IH #K0 #V0 #HLK0 #V1 #HV01 #T1 #HVT1 -#KX #VX #H #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02 -lapply (ldrop_mono … H … HLK0) -H #H destruct -elim (lpqs_ldrop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1 -elim (lpqs_inv_pair1 … H1) -H1 #K1 #W1 #HK01 #_ #H destruct -lapply (ldrop_fwd_ldrop2 … HLK1) -W1 #HLK1 -elim (lpqs_ldrop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2 -elim (lpqs_inv_pair1 … H2) -H2 #K2 #W2 #HK02 #_ #H destruct -lapply (ldrop_fwd_ldrop2 … HLK2) -W2 #HLK2 -lapply (ldrop_pair2_fwd_fw … HLK0 (#i)) -HLK0 #HLK0 -elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2 -elim (lift_total V 0 (i+1)) #T #HVT -lapply (cpqs_lift … HV1 … HLK1 … HVT1 … HVT) -K1 -V1 -lapply (cpqs_lift … HV2 … HLK2 … HVT2 … HVT) -K2 -V2 -V /2 width=3/ -qed-. - -fact cpqs_conf_lpqs_bind_bind: - ∀a,I,L0,V0,T0. ( - ∀L,T.♯{L,T} < ♯{L0,ⓑ{a,I}V0.T0} → - ∀T1. L ⊢ T ➤* T1 → ∀T2. L ⊢ T ➤* T2 → - ∀L1. L ⊢ ➤* L1 → ∀L2. L ⊢ ➤* L2 → - ∃∃T0. L1 ⊢ T1 ➤* T0 & L2 ⊢ T2 ➤* T0 - ) → - ∀V1. L0 ⊢ V0 ➤* V1 → ∀T1. L0.ⓑ{I}V0 ⊢ T0 ➤* T1 → - ∀V2. L0 ⊢ V0 ➤* V2 → ∀T2. L0.ⓑ{I}V0 ⊢ T0 ➤* T2 → - ∀L1. L0 ⊢ ➤* L1 → ∀L2. L0 ⊢ ➤* L2 → - ∃∃T. L1 ⊢ ⓑ{a,I}V1.T1 ➤* T & L2 ⊢ ⓑ{a,I}V2.T2 ➤* T. -#a #I #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01 -#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02 -elim (IH … HV01 … HV02 … HL01 … HL02) // -elim (IH … HT01 … HT02 (L1.ⓑ{I}V1) … (L2.ⓑ{I}V2)) -IH // /2 width=1/ /3 width=5/ -qed-. - -fact cpqs_conf_lpqs_bind_zeta: - ∀L0,V0,T0. ( - ∀L,T.♯{L,T} < ♯{L0,+ⓓV0.T0} → - ∀T1. L ⊢ T ➤* T1 → ∀T2. L ⊢ T ➤* T2 → - ∀L1. L ⊢ ➤* L1 → ∀L2. L ⊢ ➤* L2 → - ∃∃T0. L1 ⊢ T1 ➤* T0 & L2 ⊢ T2 ➤* T0 - ) → - ∀V1. L0 ⊢ V0 ➤* V1 → ∀T1. L0.ⓓV0 ⊢ T0 ➤* T1 → - ∀T2. L0.ⓓV0 ⊢ T0 ➤* T2 → ∀X2. ⇧[O, 1] X2 ≡ T2 → - ∀L1. L0 ⊢ ➤* L1 → ∀L2. L0 ⊢ ➤* L2 → - ∃∃T. L1 ⊢ +ⓓV1.T1 ➤* T & L2 ⊢ X2 ➤* T. -#L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01 -#T2 #HT02 #X2 #HXT2 #L1 #HL01 #L2 #HL02 -elim (IH … HT01 … HT02 (L1.ⓓV1) … (L2.ⓓV1)) -IH -HT01 -HT02 // /2 width=1/ -L0 -V0 -T0 #T #HT1 #HT2 -elim (cpqs_inv_lift1 … HT2 L2 … HXT2) -T2 /2 width=1/ /3 width=3/ -qed-. - -fact cpqs_conf_lpqs_zeta_zeta: - ∀L0,V0,T0. ( - ∀L,T.♯{L,T} < ♯{L0,+ⓓV0.T0} → - ∀T1. L ⊢ T ➤* T1 → ∀T2. L ⊢ T ➤* T2 → - ∀L1. L ⊢ ➤* L1 → ∀L2. L ⊢ ➤* L2 → - ∃∃T0. L1 ⊢ T1 ➤* T0 & L2 ⊢ T2 ➤* T0 - ) → - ∀T1. L0.ⓓV0 ⊢ T0 ➤* T1 → ∀X1. ⇧[O, 1] X1 ≡ T1 → - ∀T2. L0.ⓓV0 ⊢ T0 ➤* T2 → ∀X2. ⇧[O, 1] X2 ≡ T2 → - ∀L1. L0 ⊢ ➤* L1 → ∀L2. L0 ⊢ ➤* L2 → - ∃∃T. L1 ⊢ X1 ➤* T & L2 ⊢ X2 ➤* T. -#L0 #V0 #T0 #IH #T1 #HT01 #X1 #HXT1 -#T2 #HT02 #X2 #HXT2 #L1 #HL01 #L2 #HL02 -elim (IH … HT01 … HT02 (L1.ⓓV0) … (L2.ⓓV0)) -IH -HT01 -HT02 // /2 width=1/ -L0 -T0 #T #HT1 #HT2 -elim (cpqs_inv_lift1 … HT1 L1 … HXT1) -T1 /2 width=1/ #T1 #HT1 #HXT1 -elim (cpqs_inv_lift1 … HT2 L2 … HXT2) -T2 /2 width=1/ #T2 #HT2 #HXT2 -lapply (lift_inj … HT2 … HT1) -T #H destruct /2 width=3/ -qed-. - -fact cpqs_conf_lpqs_flat_flat: - ∀I,L0,V0,T0. ( - ∀L,T.♯{L,T} < ♯{L0,ⓕ{I}V0.T0} → - ∀T1. L ⊢ T ➤* T1 → ∀T2. L ⊢ T ➤* T2 → - ∀L1. L ⊢ ➤* L1 → ∀L2. L ⊢ ➤* L2 → - ∃∃T0. L1 ⊢ T1 ➤* T0 & L2 ⊢ T2 ➤* T0 - ) → - ∀V1. L0 ⊢ V0 ➤* V1 → ∀T1. L0 ⊢ T0 ➤* T1 → - ∀V2. L0 ⊢ V0 ➤* V2 → ∀T2. L0 ⊢ T0 ➤* T2 → - ∀L1. L0 ⊢ ➤* L1 → ∀L2. L0 ⊢ ➤* L2 → - ∃∃T. L1 ⊢ ⓕ{I}V1.T1 ➤* T & L2 ⊢ ⓕ{I}V2.T2 ➤* T. -#I #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01 -#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02 -elim (IH … HV01 … HV02 … HL01 … HL02) // -elim (IH … HT01 … HT02 … HL01 … HL02) // /3 width=5/ -qed-. - -fact cpqs_conf_lpqs_flat_tau: - ∀L0,V0,T0. ( - ∀L,T.♯{L,T} < ♯{L0,ⓝV0.T0} → - ∀T1. L ⊢ T ➤* T1 → ∀T2. L ⊢ T ➤* T2 → - ∀L1. L ⊢ ➤* L1 → ∀L2. L ⊢ ➤* L2 → - ∃∃T0. L1 ⊢ T1 ➤* T0 & L2 ⊢ T2 ➤* T0 - ) → - ∀V1,T1. L0 ⊢ T0 ➤* T1 → ∀T2. L0 ⊢ T0 ➤* T2 → - ∀L1. L0 ⊢ ➤* L1 → ∀L2. L0 ⊢ ➤* L2 → - ∃∃T. L1 ⊢ ⓝV1.T1 ➤* T & L2 ⊢ T2 ➤* T. -#L0 #V0 #T0 #IH #V1 #T1 #HT01 -#T2 #HT02 #L1 #HL01 #L2 #HL02 -elim (IH … HT01 … HT02 … HL01 … HL02) // -L0 -V0 -T0 /3 width=3/ -qed-. - -fact cpqs_conf_lpqs_tau_tau: - ∀L0,V0,T0. ( - ∀L,T.♯{L,T} < ♯{L0,ⓝV0.T0} → - ∀T1. L ⊢ T ➤* T1 → ∀T2. L ⊢ T ➤* T2 → - ∀L1. L ⊢ ➤* L1 → ∀L2. L ⊢ ➤* L2 → - ∃∃T0. L1 ⊢ T1 ➤* T0 & L2 ⊢ T2 ➤* T0 - ) → - ∀T1. L0 ⊢ T0 ➤* T1 → ∀T2. L0 ⊢ T0 ➤* T2 → - ∀L1. L0 ⊢ ➤* L1 → ∀L2. L0 ⊢ ➤* L2 → - ∃∃T. L1 ⊢ T1 ➤* T & L2 ⊢ T2 ➤* T. -#L0 #V0 #T0 #IH #T1 #HT01 -#T2 #HT02 #L1 #HL01 #L2 #HL02 -elim (IH … HT01 … HT02 … HL01 … HL02) // -L0 -V0 -T0 /2 width=3/ -qed-. - -theorem cpqs_conf_lpqs: lpx_sn_confluent cpqs cpqs. -#L0 #T0 @(f2_ind … fw … L0 T0) -L0 -T0 #n #IH #L0 * [|*] -[ #I0 #Hn #T1 #H1 #T2 #H2 #L1 #HL01 #L2 #HL02 destruct - elim (cpqs_inv_atom1 … H1) -H1 - elim (cpqs_inv_atom1 … H2) -H2 - [ #H2 #H1 destruct - /2 width=1 by cpqs_conf_lpqs_atom_atom/ - | * #K0 #V0 #V2 #i2 #HLK0 #HV02 #HVT2 #H2 #H1 destruct - /3 width=10 by cpqs_conf_lpqs_atom_delta/ - | #H2 * #K0 #V0 #V1 #i1 #HLK0 #HV01 #HVT1 #H1 destruct - /4 width=10 by ex2_commute, cpqs_conf_lpqs_atom_delta/ - | * #X #Y #V2 #z #H #HV02 #HVT2 #H2 - * #K0 #V0 #V1 #i #HLK0 #HV01 #HVT1 #H1 destruct - /3 width=17 by cpqs_conf_lpqs_delta_delta/ - ] -| #a #I #V0 #T0 #Hn #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct - elim (cpqs_inv_bind1 … H1) -H1 * - [ #V1 #T1 #HV01 #HT01 #H1 - | #T1 #HT01 #HXT1 #H11 #H12 - ] - elim (cpqs_inv_bind1 … H2) -H2 * - [1,3: #V2 #T2 #HV02 #HT02 #H2 - |2,4: #T2 #HT02 #HXT2 #H21 #H22 - ] destruct - [ /3 width=10 by cpqs_conf_lpqs_bind_bind/ - | /4 width=11 by ex2_commute, cpqs_conf_lpqs_bind_zeta/ - | /3 width=11 by cpqs_conf_lpqs_bind_zeta/ - | /3 width=12 by cpqs_conf_lpqs_zeta_zeta/ - ] -| #I #V0 #T0 #Hn #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct - elim (cpqs_inv_flat1 … H1) -H1 * - [ #V1 #T1 #HV01 #HT01 #H1 - | #HX1 #H1 - ] - elim (cpqs_inv_flat1 … H2) -H2 * - [1,3: #V2 #T2 #HV02 #HT02 #H2 - |2,4: #HX2 #H2 - ] destruct - [ /3 width=10 by cpqs_conf_lpqs_flat_flat/ - | /4 width=8 by ex2_commute, cpqs_conf_lpqs_flat_tau/ - | /3 width=8 by cpqs_conf_lpqs_flat_tau/ - | /3 width=7 by cpqs_conf_lpqs_tau_tau/ - ] -] -qed-. - -theorem cpqs_conf: ∀L. confluent … (cpqs L). -/2 width=6 by cpqs_conf_lpqs/ qed-. - theorem cpqs_trans_lpqs: lpx_sn_transitive cpqs cpqs. #L1 #T1 @(f2_ind … fw … L1 T1) -L1 -T1 #n #IH #L1 * [|*] [ #I #Hn #T #H1 #L2 #HL12 #T2 #HT2 destruct @@ -271,28 +65,6 @@ theorem cpqs_trans: ∀L. Transitive … (cpqs L). (* Properties on context-sensitive rest. parallel computation for terms *****) -lemma lpqs_cpqs_conf_dx: ∀L0,T0,T1. L0 ⊢ T0 ➤* T1 → ∀L1. L0 ⊢ ➤* L1 → - ∃∃T. L1 ⊢ T0 ➤* T & L1 ⊢ T1 ➤* T. -#L0 #T0 #T1 #HT01 #L1 #HL01 -elim (cpqs_conf_lpqs … HT01 T0 … HL01 … HL01) // -L0 /2 width=3/ -qed-. - -lemma lpqs_cpqs_conf_sn: ∀L0,T0,T1. L0 ⊢ T0 ➤* T1 → ∀L1. L0 ⊢ ➤* L1 → - ∃∃T. L1 ⊢ T0 ➤* T & L0 ⊢ T1 ➤* T. -#L0 #T0 #T1 #HT01 #L1 #HL01 -elim (cpqs_conf_lpqs … HT01 T0 … L0 … HL01) // -HT01 -HL01 /2 width=3/ -qed-. - lemma lpqs_cpqs_trans: ∀L1,L2. L1 ⊢ ➤* L2 → ∀T1,T2. L2 ⊢ T1 ➤* T2 → L1 ⊢ T1 ➤* T2. /2 width=5 by cpqs_trans_lpqs/ qed-. - -lemma fsup_cpqs_trans: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ → ∀U2. L2 ⊢ T2 ➤* U2 → - ∃∃L,U1. L1 ⊢ ➤* L & L ⊢ T1 ➤* U1 & ⦃L, U1⦄ ⊃ ⦃L2, U2⦄. -#L1 #L2 #T1 #T2 #H elim H -L1 -L2 -T1 -T2 [1,2,3,4,5: /3 width=5/ ] -#L1 #K1 #K2 #T1 #T2 #U1 #d #e #HLK1 #HTU1 #_ #IHT12 #U2 #HTU2 -elim (IHT12 … HTU2) -IHT12 -HTU2 #K #T #HK1 #HT1 #HT2 -elim (lift_total T d e) #U #HTU -elim (ldrop_lpqs_trans … HLK1 … HK1) -HLK1 -HK1 #L2 #HL12 #HL2K -lapply (cpqs_lift … HT1 … HL2K … HTU1 … HTU) -HT1 -HTU1 /3 width=11/ -qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/unfold/lpqs_ldrop.ma b/matita/matita/contribs/lambdadelta/basic_2/unfold/lpqs_ldrop.ma index bea4a9134..2b838d285 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/unfold/lpqs_ldrop.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/unfold/lpqs_ldrop.ma @@ -12,6 +12,7 @@ (* *) (**************************************************************************) +include "basic_2/relocation/fsup.ma". include "basic_2/relocation/ldrop_lpx_sn.ma". include "basic_2/unfold/cpqs_lift.ma". include "basic_2/unfold/lpqs.ma". @@ -28,3 +29,15 @@ lemma ldrop_lpqs_trans: dedropable_sn lpqs. lemma lpqs_ldrop_trans_O1: dropable_dx lpqs. /2 width=3 by lpx_sn_dropable/ qed-. + +(* Properties on context-sensitive rest. parallel computation for terms *****) + +lemma fsup_cpqs_trans: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ → ∀U2. L2 ⊢ T2 ➤* U2 → + ∃∃L,U1. L1 ⊢ ➤* L & L ⊢ T1 ➤* U1 & ⦃L, U1⦄ ⊃ ⦃L2, U2⦄. +#L1 #L2 #T1 #T2 #H elim H -L1 -L2 -T1 -T2 [1,2,3,4,5: /3 width=5/ ] +#L1 #K1 #K2 #T1 #T2 #U1 #d #e #HLK1 #HTU1 #_ #IHT12 #U2 #HTU2 +elim (IHT12 … HTU2) -IHT12 -HTU2 #K #T #HK1 #HT1 #HT2 +elim (lift_total T d e) #U #HTU +elim (ldrop_lpqs_trans … HLK1 … HK1) -HLK1 -HK1 #L2 #HL12 #HL2K +lapply (cpqs_lift … HT1 … HL2K … HTU1 … HTU) -HT1 -HTU1 /3 width=11/ +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/unfold/lpqs_lpqs.ma b/matita/matita/contribs/lambdadelta/basic_2/unfold/lpqs_lpqs.ma index c134c8880..566dd5347 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/unfold/lpqs_lpqs.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/unfold/lpqs_lpqs.ma @@ -16,6 +16,227 @@ include "basic_2/unfold/lpqs_cpqs.ma". (* SN RESTRICTED PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS *****************) +(* Main properties on context-sensitive rest parallel computation for terms *) + +fact cpqs_conf_lpqs_atom_atom: + ∀I,L1,L2. ∃∃T. L1 ⊢ ⓪{I} ➤* T & L2 ⊢ ⓪{I} ➤* T. +/2 width=3/ qed-. + +fact cpqs_conf_lpqs_atom_delta: + ∀L0,i. ( + ∀L,T.♯{L, T} < ♯{L0, #i} → + ∀T1. L ⊢ T ➤* T1 → ∀T2. L ⊢ T ➤* T2 → + ∀L1. L ⊢ ➤* L1 → ∀L2. L ⊢ ➤* L2 → + ∃∃T0. L1 ⊢ T1 ➤* T0 & L2 ⊢ T2 ➤* T0 + ) → + ∀K0,V0. ⇩[O, i] L0 ≡ K0.ⓓV0 → + ∀V2. K0 ⊢ V0 ➤* V2 → ∀T2. ⇧[O, i + 1] V2 ≡ T2 → + ∀L1. L0 ⊢ ➤* L1 → ∀L2. L0 ⊢ ➤* L2 → + ∃∃T. L1 ⊢ #i ➤* T & L2 ⊢ T2 ➤* T. +#L0 #i #IH #K0 #V0 #HLK0 #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02 +elim (lpqs_ldrop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1 +elim (lpqs_inv_pair1 … H1) -H1 #K1 #V1 #HK01 #HV01 #H destruct +elim (lpqs_ldrop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2 +elim (lpqs_inv_pair1 … H2) -H2 #K2 #W2 #HK02 #_ #H destruct +lapply (ldrop_fwd_ldrop2 … HLK2) -W2 #HLK2 +lapply (ldrop_pair2_fwd_fw … HLK0 (#i)) -HLK0 #HLK0 +elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2 +elim (lift_total V 0 (i+1)) #T #HVT +lapply (cpqs_lift … HV2 … HLK2 … HVT2 … HVT) -K2 -V2 /3 width=6/ +qed-. + +fact cpqs_conf_lpqs_delta_delta: + ∀L0,i. ( + ∀L,T.♯{L, T} < ♯{L0, #i} → + ∀T1. L ⊢ T ➤* T1 → ∀T2. L ⊢ T ➤* T2 → + ∀L1. L ⊢ ➤* L1 → ∀L2. L ⊢ ➤* L2 → + ∃∃T0. L1 ⊢ T1 ➤* T0 & L2 ⊢ T2 ➤* T0 + ) → + ∀K0,V0. ⇩[O, i] L0 ≡ K0.ⓓV0 → + ∀V1. K0 ⊢ V0 ➤* V1 → ∀T1. ⇧[O, i + 1] V1 ≡ T1 → + ∀KX,VX. ⇩[O, i] L0 ≡ KX.ⓓVX → + ∀V2. KX ⊢ VX ➤* V2 → ∀T2. ⇧[O, i + 1] V2 ≡ T2 → + ∀L1. L0 ⊢ ➤* L1 → ∀L2. L0 ⊢ ➤* L2 → + ∃∃T. L1 ⊢ T1 ➤* T & L2 ⊢ T2 ➤* T. +#L0 #i #IH #K0 #V0 #HLK0 #V1 #HV01 #T1 #HVT1 +#KX #VX #H #V2 #HV02 #T2 #HVT2 #L1 #HL01 #L2 #HL02 +lapply (ldrop_mono … H … HLK0) -H #H destruct +elim (lpqs_ldrop_conf … HLK0 … HL01) -HL01 #X1 #H1 #HLK1 +elim (lpqs_inv_pair1 … H1) -H1 #K1 #W1 #HK01 #_ #H destruct +lapply (ldrop_fwd_ldrop2 … HLK1) -W1 #HLK1 +elim (lpqs_ldrop_conf … HLK0 … HL02) -HL02 #X2 #H2 #HLK2 +elim (lpqs_inv_pair1 … H2) -H2 #K2 #W2 #HK02 #_ #H destruct +lapply (ldrop_fwd_ldrop2 … HLK2) -W2 #HLK2 +lapply (ldrop_pair2_fwd_fw … HLK0 (#i)) -HLK0 #HLK0 +elim (IH … HLK0 … HV01 … HV02 … HK01 … HK02) -L0 -K0 -V0 #V #HV1 #HV2 +elim (lift_total V 0 (i+1)) #T #HVT +lapply (cpqs_lift … HV1 … HLK1 … HVT1 … HVT) -K1 -V1 +lapply (cpqs_lift … HV2 … HLK2 … HVT2 … HVT) -K2 -V2 -V /2 width=3/ +qed-. + +fact cpqs_conf_lpqs_bind_bind: + ∀a,I,L0,V0,T0. ( + ∀L,T.♯{L,T} < ♯{L0,ⓑ{a,I}V0.T0} → + ∀T1. L ⊢ T ➤* T1 → ∀T2. L ⊢ T ➤* T2 → + ∀L1. L ⊢ ➤* L1 → ∀L2. L ⊢ ➤* L2 → + ∃∃T0. L1 ⊢ T1 ➤* T0 & L2 ⊢ T2 ➤* T0 + ) → + ∀V1. L0 ⊢ V0 ➤* V1 → ∀T1. L0.ⓑ{I}V0 ⊢ T0 ➤* T1 → + ∀V2. L0 ⊢ V0 ➤* V2 → ∀T2. L0.ⓑ{I}V0 ⊢ T0 ➤* T2 → + ∀L1. L0 ⊢ ➤* L1 → ∀L2. L0 ⊢ ➤* L2 → + ∃∃T. L1 ⊢ ⓑ{a,I}V1.T1 ➤* T & L2 ⊢ ⓑ{a,I}V2.T2 ➤* T. +#a #I #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01 +#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02 +elim (IH … HV01 … HV02 … HL01 … HL02) // +elim (IH … HT01 … HT02 (L1.ⓑ{I}V1) … (L2.ⓑ{I}V2)) -IH // /2 width=1/ /3 width=5/ +qed-. + +fact cpqs_conf_lpqs_bind_zeta: + ∀L0,V0,T0. ( + ∀L,T.♯{L,T} < ♯{L0,+ⓓV0.T0} → + ∀T1. L ⊢ T ➤* T1 → ∀T2. L ⊢ T ➤* T2 → + ∀L1. L ⊢ ➤* L1 → ∀L2. L ⊢ ➤* L2 → + ∃∃T0. L1 ⊢ T1 ➤* T0 & L2 ⊢ T2 ➤* T0 + ) → + ∀V1. L0 ⊢ V0 ➤* V1 → ∀T1. L0.ⓓV0 ⊢ T0 ➤* T1 → + ∀T2. L0.ⓓV0 ⊢ T0 ➤* T2 → ∀X2. ⇧[O, 1] X2 ≡ T2 → + ∀L1. L0 ⊢ ➤* L1 → ∀L2. L0 ⊢ ➤* L2 → + ∃∃T. L1 ⊢ +ⓓV1.T1 ➤* T & L2 ⊢ X2 ➤* T. +#L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01 +#T2 #HT02 #X2 #HXT2 #L1 #HL01 #L2 #HL02 +elim (IH … HT01 … HT02 (L1.ⓓV1) … (L2.ⓓV1)) -IH -HT01 -HT02 // /2 width=1/ -L0 -V0 -T0 #T #HT1 #HT2 +elim (cpqs_inv_lift1 … HT2 L2 … HXT2) -T2 /2 width=1/ /3 width=3/ +qed-. + +fact cpqs_conf_lpqs_zeta_zeta: + ∀L0,V0,T0. ( + ∀L,T.♯{L,T} < ♯{L0,+ⓓV0.T0} → + ∀T1. L ⊢ T ➤* T1 → ∀T2. L ⊢ T ➤* T2 → + ∀L1. L ⊢ ➤* L1 → ∀L2. L ⊢ ➤* L2 → + ∃∃T0. L1 ⊢ T1 ➤* T0 & L2 ⊢ T2 ➤* T0 + ) → + ∀T1. L0.ⓓV0 ⊢ T0 ➤* T1 → ∀X1. ⇧[O, 1] X1 ≡ T1 → + ∀T2. L0.ⓓV0 ⊢ T0 ➤* T2 → ∀X2. ⇧[O, 1] X2 ≡ T2 → + ∀L1. L0 ⊢ ➤* L1 → ∀L2. L0 ⊢ ➤* L2 → + ∃∃T. L1 ⊢ X1 ➤* T & L2 ⊢ X2 ➤* T. +#L0 #V0 #T0 #IH #T1 #HT01 #X1 #HXT1 +#T2 #HT02 #X2 #HXT2 #L1 #HL01 #L2 #HL02 +elim (IH … HT01 … HT02 (L1.ⓓV0) … (L2.ⓓV0)) -IH -HT01 -HT02 // /2 width=1/ -L0 -T0 #T #HT1 #HT2 +elim (cpqs_inv_lift1 … HT1 L1 … HXT1) -T1 /2 width=1/ #T1 #HT1 #HXT1 +elim (cpqs_inv_lift1 … HT2 L2 … HXT2) -T2 /2 width=1/ #T2 #HT2 #HXT2 +lapply (lift_inj … HT2 … HT1) -T #H destruct /2 width=3/ +qed-. + +fact cpqs_conf_lpqs_flat_flat: + ∀I,L0,V0,T0. ( + ∀L,T.♯{L,T} < ♯{L0,ⓕ{I}V0.T0} → + ∀T1. L ⊢ T ➤* T1 → ∀T2. L ⊢ T ➤* T2 → + ∀L1. L ⊢ ➤* L1 → ∀L2. L ⊢ ➤* L2 → + ∃∃T0. L1 ⊢ T1 ➤* T0 & L2 ⊢ T2 ➤* T0 + ) → + ∀V1. L0 ⊢ V0 ➤* V1 → ∀T1. L0 ⊢ T0 ➤* T1 → + ∀V2. L0 ⊢ V0 ➤* V2 → ∀T2. L0 ⊢ T0 ➤* T2 → + ∀L1. L0 ⊢ ➤* L1 → ∀L2. L0 ⊢ ➤* L2 → + ∃∃T. L1 ⊢ ⓕ{I}V1.T1 ➤* T & L2 ⊢ ⓕ{I}V2.T2 ➤* T. +#I #L0 #V0 #T0 #IH #V1 #HV01 #T1 #HT01 +#V2 #HV02 #T2 #HT02 #L1 #HL01 #L2 #HL02 +elim (IH … HV01 … HV02 … HL01 … HL02) // +elim (IH … HT01 … HT02 … HL01 … HL02) // /3 width=5/ +qed-. + +fact cpqs_conf_lpqs_flat_tau: + ∀L0,V0,T0. ( + ∀L,T.♯{L,T} < ♯{L0,ⓝV0.T0} → + ∀T1. L ⊢ T ➤* T1 → ∀T2. L ⊢ T ➤* T2 → + ∀L1. L ⊢ ➤* L1 → ∀L2. L ⊢ ➤* L2 → + ∃∃T0. L1 ⊢ T1 ➤* T0 & L2 ⊢ T2 ➤* T0 + ) → + ∀V1,T1. L0 ⊢ T0 ➤* T1 → ∀T2. L0 ⊢ T0 ➤* T2 → + ∀L1. L0 ⊢ ➤* L1 → ∀L2. L0 ⊢ ➤* L2 → + ∃∃T. L1 ⊢ ⓝV1.T1 ➤* T & L2 ⊢ T2 ➤* T. +#L0 #V0 #T0 #IH #V1 #T1 #HT01 +#T2 #HT02 #L1 #HL01 #L2 #HL02 +elim (IH … HT01 … HT02 … HL01 … HL02) // -L0 -V0 -T0 /3 width=3/ +qed-. + +fact cpqs_conf_lpqs_tau_tau: + ∀L0,V0,T0. ( + ∀L,T.♯{L,T} < ♯{L0,ⓝV0.T0} → + ∀T1. L ⊢ T ➤* T1 → ∀T2. L ⊢ T ➤* T2 → + ∀L1. L ⊢ ➤* L1 → ∀L2. L ⊢ ➤* L2 → + ∃∃T0. L1 ⊢ T1 ➤* T0 & L2 ⊢ T2 ➤* T0 + ) → + ∀T1. L0 ⊢ T0 ➤* T1 → ∀T2. L0 ⊢ T0 ➤* T2 → + ∀L1. L0 ⊢ ➤* L1 → ∀L2. L0 ⊢ ➤* L2 → + ∃∃T. L1 ⊢ T1 ➤* T & L2 ⊢ T2 ➤* T. +#L0 #V0 #T0 #IH #T1 #HT01 +#T2 #HT02 #L1 #HL01 #L2 #HL02 +elim (IH … HT01 … HT02 … HL01 … HL02) // -L0 -V0 -T0 /2 width=3/ +qed-. + +theorem cpqs_conf_lpqs: lpx_sn_confluent cpqs cpqs. +#L0 #T0 @(f2_ind … fw … L0 T0) -L0 -T0 #n #IH #L0 * [|*] +[ #I0 #Hn #T1 #H1 #T2 #H2 #L1 #HL01 #L2 #HL02 destruct + elim (cpqs_inv_atom1 … H1) -H1 + elim (cpqs_inv_atom1 … H2) -H2 + [ #H2 #H1 destruct + /2 width=1 by cpqs_conf_lpqs_atom_atom/ + | * #K0 #V0 #V2 #i2 #HLK0 #HV02 #HVT2 #H2 #H1 destruct + /3 width=10 by cpqs_conf_lpqs_atom_delta/ + | #H2 * #K0 #V0 #V1 #i1 #HLK0 #HV01 #HVT1 #H1 destruct + /4 width=10 by ex2_commute, cpqs_conf_lpqs_atom_delta/ + | * #X #Y #V2 #z #H #HV02 #HVT2 #H2 + * #K0 #V0 #V1 #i #HLK0 #HV01 #HVT1 #H1 destruct + /3 width=17 by cpqs_conf_lpqs_delta_delta/ + ] +| #a #I #V0 #T0 #Hn #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct + elim (cpqs_inv_bind1 … H1) -H1 * + [ #V1 #T1 #HV01 #HT01 #H1 + | #T1 #HT01 #HXT1 #H11 #H12 + ] + elim (cpqs_inv_bind1 … H2) -H2 * + [1,3: #V2 #T2 #HV02 #HT02 #H2 + |2,4: #T2 #HT02 #HXT2 #H21 #H22 + ] destruct + [ /3 width=10 by cpqs_conf_lpqs_bind_bind/ + | /4 width=11 by ex2_commute, cpqs_conf_lpqs_bind_zeta/ + | /3 width=11 by cpqs_conf_lpqs_bind_zeta/ + | /3 width=12 by cpqs_conf_lpqs_zeta_zeta/ + ] +| #I #V0 #T0 #Hn #X1 #H1 #X2 #H2 #L1 #HL01 #L2 #HL02 destruct + elim (cpqs_inv_flat1 … H1) -H1 * + [ #V1 #T1 #HV01 #HT01 #H1 + | #HX1 #H1 + ] + elim (cpqs_inv_flat1 … H2) -H2 * + [1,3: #V2 #T2 #HV02 #HT02 #H2 + |2,4: #HX2 #H2 + ] destruct + [ /3 width=10 by cpqs_conf_lpqs_flat_flat/ + | /4 width=8 by ex2_commute, cpqs_conf_lpqs_flat_tau/ + | /3 width=8 by cpqs_conf_lpqs_flat_tau/ + | /3 width=7 by cpqs_conf_lpqs_tau_tau/ + ] +] +qed-. + +theorem cpqs_conf: ∀L. confluent … (cpqs L). +/2 width=6 by cpqs_conf_lpqs/ qed-. + +(* Properties on context-sensitive rest. parallel computation for terms *****) + +lemma lpqs_cpqs_conf_dx: ∀L0,T0,T1. L0 ⊢ T0 ➤* T1 → ∀L1. L0 ⊢ ➤* L1 → + ∃∃T. L1 ⊢ T0 ➤* T & L1 ⊢ T1 ➤* T. +#L0 #T0 #T1 #HT01 #L1 #HL01 +elim (cpqs_conf_lpqs … HT01 T0 … HL01 … HL01) // -L0 /2 width=3/ +qed-. + +lemma lpqs_cpqs_conf_sn: ∀L0,T0,T1. L0 ⊢ T0 ➤* T1 → ∀L1. L0 ⊢ ➤* L1 → + ∃∃T. L1 ⊢ T0 ➤* T & L0 ⊢ T1 ➤* T. +#L0 #T0 #T1 #HT01 #L1 #HL01 +elim (cpqs_conf_lpqs … HT01 T0 … L0 … HL01) // -HT01 -HL01 /2 width=3/ +qed-. + (* Main properties **********************************************************) theorem lpqs_conf: confluent … lpqs. diff --git a/matita/matita/contribs/lambdadelta/basic_2/web/basic_2_src.tbl b/matita/matita/contribs/lambdadelta/basic_2/web/basic_2_src.tbl index a2d504130..a9ccf6682 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/web/basic_2_src.tbl +++ b/matita/matita/contribs/lambdadelta/basic_2/web/basic_2_src.tbl @@ -101,7 +101,7 @@ table { ] [ { "context-sensitive computation" * } { [ "lprs ( ? ⊢ ➡* ? )" "lprs_alt ( ? ⊢ ➡➡* ? )" "lprs_ldrop" + "lprs_aaa" + "lprs_cprs" + "lprs_lprs" * ] - [ "cprs ( ? ⊢ ? ➡* ?)" "cprs_lift" + "cprs_tpss" + "cprs_ltpss_dx" + "cprs_ltpss_sn" + "cprs_aaa" + "cprs_lpr" + "cprs_cprs" + "cprs_tstc" + "cprs_tstc_vector" * ] + [ "cprs ( ? ⊢ ? ➡* ?)" "cprs_lift" + "cprs_cpss" + "cprs_ltpss_dx" + "cprs_ltpss_sn" + "cprs_aaa" + "cprs_lpr" + "cprs_cprs" + "cprs_tstc" + "cprs_tstc_vector" * ] } ] [ { "local env. ref. for abstract candidates of reducibility" * } { @@ -121,7 +121,7 @@ table { } ] [ { "context-sensitive reduction" * } { - [ "lpr ( ? ⊢ ➡ ? )" "lpr_ldrop" + "lpr_cpss" + "lpr_lpss" + "lpr_aaa" + "lpr_cpr" + "lpr_lpr" * ] + [ "lpr ( ? ⊢ ➡ ? )" "lpr_ldrop" + "lpr_lpss" + "lpr_aaa" + "lpr_lpr" * ] [ "cpr ( ? ⊢ ? ➡ ? )" "cpr_lift" + "cpr_cif" * ] } ] -- 2.39.2